BOUNDARY COHOMOLOGY OF SHIMURA VARIETIES, III:

Coherent cohomology on higher-rank boundary strata and applications to Hodge theory

Michael Harris^{1}

UFR de Mathématique, Université Paris 7, 2 Pl. Jussieu 75251 Paris cedex 05, FRANCE

Steven Zucker^{2}

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218 USA

Introduction

The present article continues the study of the boundary cohomology of Shimura varieties initiated in [HZ1, HZ2]. Let G be a reductive group over ℚ, X the symmetric space associated to G(ℝ), and Γ an arithmetic subgroup (e.g., a congruence subgroup) of G(ℚ). We consider the cohomology of Γ\X with coefficients in the local system Ṽ constructed from a representation V of G, i.e., H^{•}(Γ\X, Ṽ) ≃ H^{•}(Γ, V). It is standard that this cohomology can be decomposed as the direct sum of “interior” cohomology, defined as the image of the cohomology with compact supports H_{c}^{•}(Γ\X, Ṽ), and a complementary “boundary cohomology” that restricts nontrivially to the boundary of the Borel-Serre (manifold-with-corners) compactification of Γ\X. The designation of boundary cohomology is generally non-canonical, and much work has been devoted to constructing canonical decompositions using Eisenstein series.

By an elaboration on the de Rham theorem, one knows that the cohomology group H^{•}(Γ\X, Ṽ) can be expressed as the relative Lie algebra cohomology of the space of V -valued C^{∞} functions on Γ\G(ℝ), or even the functions of moderate growth ([B2, §7]). Thanks to the work of Franke [Fr1], one can replace the functions of moderate growth by the subspace of automorphic forms, and this can provide the starting point for an approach to the boundary cohomology. However, in this series of articles we are concerned only tangentially with the relation between boundary cohomology and automorphic forms. We choose to work at a more intrinsic level, concentrating instead on the additional structures on H^{•}(Γ\X, Ṽ) when X is a hermitian symmetric domain. In that case, Γ\X is an algebraic variety, and Ṽ

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1Institut de Mathématiques de Jussieu, U.M.R. 7586 du CNRS, Supported in part by the National Science Foundation, through Grant DMS-9203142

2Supported in part by the National Science Foundation, through Grants DMS9423689 and DMS9820958

underlies a natural variation of Hodge structure. Morihiko Saito’s theory of mixed Hodge modules [Sa3] then gives that H^{•}(Γ\X, Ṽ) has a corresponding mixed Hodge structure (MHS). The nature of this MHS at the boundary—more accurately, the associated MHS on the deleted neighborhood cohomology of the boundary—was the subject of [HZ2].

The “adelic version” of Γ\X is the Shimura variety Sh(G, X), whose connected components are of the form Γ\X. This has a canonical model over a number field E. The de Rham isomorphism identifies H^{•}(Sh(G, X), Ṽ) with the hypercohomology of of an E-rational complex of coherent sheaves on Sh(G, X); thus H^{•}(Sh(G, X), Ṽ) acquires an E-rational structure distinct from the topological rational structure coming from the coefficients Ṽ. In particular, H^{•}(Sh(G, X), Ṽ) has a Hodge filtration whose graded pieces are given by the coherent cohomology with coefficients in certain automorphic vector bundles [H1, Mi2]; the latter have natural E-rational structure. (This E-rationality can be asserted for de Rham cohomology, without grading for the Hodge filtration, and that is conjecturally equivalent in this context.) Study of the boundary cohomology of such automorphic vector bundles was begun in [HZ1].

Both [HZ1] and [HZ2] made essential use of toroidal compactifications of Shimura varieties (which has its origin in [AMRT]), following [H3] and [H4]. The toroidal boundary of Sh(G, X) (a divisor with normal crossings), like the Borel-Serre boundary, is stratified according to conjugacy classes of parabolic subgroups of G. The cohomology of the boundary, both in the topological setting (as above) and in the coherent setting (i.e., for canonically extended automorphic vector bundles), can be computed as the abutment of the spectral sequence for the closed covering given by this stratification; this is called the nerve spectral sequence. In [HZ1] we analyzed the contribution to the nerve spectral sequence from the strata associated to maximal parabolics in the coherent setting. The first task of the present article is to extend this analysis to general parabolics, thereby fulfilling our promise from [HZ1], and this is carried out in the first three Sections. This necessitated a generalization in Section 1 of much of the machinery of automorphic vector bundles to the toroidal compactifications of mixed Shimura varieties (constructed by Pink [P]).

Most of the calculations from [HZ1, §3] go over without change, but there are a few delicate points, notably the issue of basechange in (3.4). For the latter, we must recall the role of conditions of growth and decay. These entered in the coherent setting when we established the existence and degeneration of Leray spectral sequences for morphisms of toroidally compactified varieties and the corresponding morphisms of canonical (or subcanonical) extensions of automorphic vector bundles. In effect, it enabled us to circumvent the complications related to basechange at infinity. As suggested above, this last point recurs here. We are obliged to prove (in (2.3)) a generalization to mixed conditions of growth and decay, enabling us, in effect, to isolate a single boundary stratum.

Our main result in Section 3 is that the differentials in the E_{1}-term of the nerve spectral sequence for coherent cohomology decompose naturally into pieces that either are given in terms of restriction maps on pure Shimura varieties or are “purely topological” (see (3.5.4)). Via Franke’s interpretation of cohomology in terms of automorphic forms, this implies (see (3.6)) that the constant term maps for cohomology, expressed as integration of an automorphic form along the unipotent radical of appropriate parabolic subgroups, are rational with respect to the de Rham rational structure; for maximal parabolics, this was already obtained in [HZ1, 4.8].

The nerve spectral sequence for the topological cohomology H^{•}(Γ\X, Ṽ) was studied in detail in [HZ2]. Hodge-theoretic considerations require (algebraic) compactifications, and the toroidal compactifications were convenient to use for this purpose as well. It was a subtle matter to compare the deleted neighborhoods of the Borel-Serre and toroidal boundary strata associated to a given parabolic subgroup (see [HZ2, §2]). We constructed isomorphisms between them that are compatible with restriction maps, allowing for transport of structure from the latter to the former. From this, it follows that the differentials in the topological nerve spectral sequence are morphisms of mixed Hodge structures. In particular, they induce maps after grading for the Hodge filtration F. Since a morphism of mixed Hodge structures is determined by its gradation for F, it follows, for instance, that ghost classes exist in H^{•}(Γ\X, Ṽ) if and only if they exist in Gr_{F} H^{•}(Γ\X, Ṽ) (see (4.6.7)). (Recall that a ghost class in H^{•}(Γ\X, Ṽ) is a cohomology class whose restriction to the Borel-Serre boundary is non-zero, yet whose restriction to each face (stratum) thereof is zero.) In (4.1), we compare the graded differentials to the results obtained for the differentials in the case of the coherent cohomology. To that end, we derive a formula for the deleted neighborhood cohomology of a boundary stratum as de Rham cohomology on a suitable toroidal compactification of the associated (Baily-Borel) boundary component (see (4.1.9)).

Of course, the above can be repeated for the weight filtration. For an example of the use of weights to rule out ghost classes (cf. (4.6.14)), see [Z5, App. A].^{3} We are still seeking a satisfactory way of dealing with the entire mixed Hodge structure. It is therefore strongly to be feared that this article is not the last of the series . . .. The content of the first three sections of this article completes the verification of results announced in [HZ1, §5] and in [H5]. They can be summarized by saying that the (topological) nerve spectral sequence is a spectral sequence of mixed Hodge-de Rham structures over the field of definition of the canonical model.

In Section 4, we continue to develop the Hodge theoretic material from [HZ2, §5]. In (4.2), we reformulate the results in (4.1) by using the “minimal model” of the holomorphic de Rham complex, viz., the dual Bernstein-Gelfand-Gelfand complex, and deduce the E-rational version of (4.1.9).

A big surprise in this work was the discovery of another interesting filtration on the boundary complex, whose spectral sequence is, like the nerve spectral sequence, a spectral sequence of mixed Hodge structures. In a way, there is nothing new about this filtration, which we call the filtration by holomorphic rank; it is given by the pullback to the Borel-Serre boundary of the filtration of the Baily-Borel Satake boundary by (unions of) boundary strata of increasing dimension (see also (4.4.15)). In a sense that can be made precise, its E_{1}-term is closer to the abutment than that of the nerve spectral sequence, though further from the question of ghost classes. We treat the holomorphic rank filtration in (4.4), though the same considerations already show up in (3.5) in the coherent setting. Cases of the latter give the Hodge components for the E_{1}-term of the topological holomorphic rank spectral sequence, and this gets examined in (4.5).

Several fundamental questions remain open. The analysis of cohomology of Shimura varieties should be extended to the intersection cohomology of their minimal (Baily-Borel) compactifications. The Zucker conjecture, proved by Looijenga [L] and Saper-Stern [SS], asserts that this cohomology is isomorphic to the L_{2}- cohomology, or again to the Lie algebra cohomology of square-integrable C^{∞} functions, or by [Fr1], of square-integrable automorphic forms. However, it is not known whether this isomorphism identifies Morihiko Saito’s Hodge structure with the analytic Hodge structure on L_{2}-cohomology (the one given a priori by the L^{2} harmonic forms). In Section 5 we obtain a partial result in this direction: we show that the map from L_{2}-cohomology of the open Shimura variety to ordinary cohomology is a morphism of (mixed) Hodge structures (this is a small improvement over what was asserted in [H5, 3.3.9]). We do not address the question of whether intersection cohomology carries a de Rham rational structure.

3The correct outcome of the calculation presented in the latter is that there are no ghosts for GSp(4) when the representation V is generic, i.e., where the highest weight for Sp(4) has positive inner product with both simple roots. When V is trivial, on the other hand, the calculation does allow for a weight-two ghost class in H^{2}(Γ\X, ℚ), and such a class is determined in [KR, 14.1.3].

It is also true that not all questions are treated in maximum generality. For instance, we have not studied the cohomology of a general automorphic vector bundle or variation of mixed Hodge structure on a mixed Shimura variety, but have rather been content to work out the cases directly relevant to the cohomology of pure Shimura varieties. Experience suggests these omissions will return to haunt us (providing even more impetus for article IV?). Another thing absent is the exploration of relations between our constructions and the general polylogarithms constructed by Wildeshaus [W1, W2].

Much of this work was begun at the time of writing of [H5], where some of our results were announced. The actual writing of the present article did not get under way until the second-named author visited Université Paris 7 in May, 1997. We both wish to thank that institution for the hospitality extended on that occasion. Likewise, a large amount of the work and writing of this article was carried out while the second-named author was spending Academic Year 1998–99 on sabbatical at the Institute for Advanced Study in Princeton. We also wish to thank P. Polo for helpful discussions of the generalized Bernstein–Bernstein– Gelfand resolution, and Z. Mebkhout for help with the proof of Proposition (4.2.21). We thank J. Wildeshaus for numerous thoughtful comments on both the content and the exposition of the article. Finally, we are grateful to the referee for his careful reading of the first version of the present article and for his numerous suggestions, which we believe have led to a substantial improvement of the first section, in particular.

Contents

Introduction Conventions and notation

1. Automorphic vector bundles on mixed Shimura varieties (1.1) Mixed Shimura varieties (1.2) Construction of automorphic vector bundles (1.3) Canonical extensions of automorphic vector bundles (1.4) Functorial properties of canonical extensions (1.5) A boundary calculation

2. Mixed growth conditions and coherent cohomology (2.1) \overline{∂} with logarithmic growth in one variable (2.2) Logarithmic growth in several variables (2.3) Forms with mixed growth conditions (2.4) Siegel sets in mixed Shimura varieties (2.5) Toroidal compactifications and Siegel sets (2.6) Growth conditions at the toroidal boundary (2.7) Mixed growth conditions and basechange

3. The nerve spectral sequence for coherent cohomology (3.1) Cohomology on boundary strata of toroidal compactifications (3.2) The nerve spectral sequence for boundary cohomology (3.3) Differentials in the nerve spectral sequence (3.4) Differentials in the nerve spectral sequence, concluded (3.5) Towards the determination of E_{2} (3.6) Differentials and automorphic forms

4. Hodge theoretic applications (4.1) The Hodge filtration for boundary cohomology (4.2) The dual Bernstein-Gelfand-Gelfand complex (4.3) Hodge theory at the boundary, revisited (4.4) The topological boundary cohomology (4.5) The mixed Hodge complex filtered by holomorphic rank (4.6) Ghost classes

5. On the comparison of Hodge structures References

Conventions and notation

(0.1) We let \underline{S} = R_{ℂ/ℝ}𝔾_{m,ℂ} be the torus whose real representations parametrize ℝ-

Hodge structures. To any real representation h : \underline{S} → GL(V) is associated a Hodge

structure on V such that V^{p,q} is the subspace of V_{ℂ} on which z ∈ \underline{S}(ℝ)⟶^{∼}ℂ^{×} acts

as z^{−p}\overline{z}^{−q}. The Hodge structure thus obtained is canonically the direct sum of its

weight graded components. We let μ : 𝔾_{m,ℂ} → \underline{S}_{ℂ} be the cocharacter such that

h ∘ μ(z) acts as z^{−p} on V^{p,q}, for any h and V as above. We let w : 𝔾_{m,ℝ} → \underline{S}

be the natural map. Then with (h, V) as above, the associated Hodge structure is

pure of weight w(V) if and only if h ∘ w defines a character of 𝔾_{m,ℝ}, and in that

case h ∘ w(t) = t^{−w(V)}.

(0.2) Let G be a connected reductive group over ℂ, P ⊂ G a parabolic subgroup,

L ⊂ P a Levi subgroup, H ⊂ L a maximal torus, R_{u}P ⊂ P the unipotent radical.

Let B ⊂ P be a Borel subgroup of G containing H. We use the lower case Gothic

letters 𝔤, 𝔭, 𝔥, 𝔩, 𝔟 to denote the corresponding Lie algebras, and let Φ = Φ(𝔤, 𝔥)

be the set of roots, Φ^{+} the positive roots corresponding to 𝔟, Φ_{L} = Φ(𝔩, 𝔥) and Φ_{L}^{+}

respectively the roots and positive roots of 𝔥 in 𝔩. Let W = W (𝔤, 𝔥) be the Weyl

group, W_{L} = W (𝔩, 𝔥). The subset W^{P} ⊂ W of Kostant representatives of W_{L}\W

is the set

{w ∈ W | w^{−1}(α) > 0, ∀α ∈ Φ_{L}^{+}}.

Then W^{P} is the set of representatives of shortest length for the right cosets W_{L}\W.

The same terminology is used for the shortest representatives of W_{L} in W when G is quasi-split over a field of characteristic zero. (0.3) Starting in §3 we will concentrate on the case of pure Shimura varieties. Notation is as in [HZ1] and [HZ2] (see, however, (0.5)). In particular, (G, X) is a Shimura datum, K_{p} ⊂ G_{ℝ} is the (algebraic) identity component of the stabilizer of a point p ∈ X, H ⊂ K_{p} is a maximal torus, 𝔥 its Lie algebra. We assume a set of positive roots Φ^{+} chosen as above, with decomposition Φ^{+} = Φ_{c}^{+} ∪ Φ_{n}^{+} into compact and non-compact roots, as in [HZ1, 3.6] (where we used the letter R instead of Φ). All references to highest weights will be with respect to these choices. Let 𝔓^{+} ⊂ 𝔤 (resp. 𝔓^{−}) be the maximal parabolic subalgebra with Levi component 𝔨_{p} and unipotent radical 𝔭^{+} (resp. 𝔭^{−}). (0.4) We wish to recall the combinatorics of the intersection of boundary strata in the Borel-Serre and toroidal compactifications of the locally symmetric variety Γ\X. This goes the same for both, so we will treat only the former.

The Borel-Serre compactification is Γ\\overline{X}, where \overline{X} is a certain manifold-with-corners with G(ℚ)-action, with X as interior. Its closed (or open) boundary faces are in natural one-to-one correspondence with the rational parabolic subgroups P of G, and are denoted \overline{e(P)}. Then \overline{e(P)} ∩ \overline{e(Q)} ≠ ∅ if and only if P ∩ Q is parabolic, and then the intersection is simply \overline{e(P ∩ Q)}.

In Γ\\overline{X}, one has boundary faces indexed by Γ-conjugacy classes of rational parabolic subgroups. It is customary to take representatives of these conjugacy classes, so one must understand the notational complication that ensues. The boundary face \overline{e′(P)} of Γ\\overline{X} is an arithmetic quotient of \overline{e(P)}, so likewise for any Γ-conjugate of P. Thus, it is wrong to expect that \overline{e′(P)} ∩ \overline{e′(Q)} = \overline{e′(P ∩ Q)} when P ∩ Q is parabolic. Indeed, \overline{e′(P ∩ Q)} is only one connected component of the intersection. Suppose that P ∩ Q is the parabolic subgroup R. Then \overline{e′(P)} ∩ \overline{e′(Q)} is a finite disjoint union of faces of the form \overline{e′(gRg^{−1})}, for g running over representatives of the fiber over the identity double coset in the right-hand side of

(Γ ∩ P)\P (ℚ)/R(ℚ) ⟶ Γ\G(ℚ)/Q(ℚ).

This understanding underlies all treatment of intersections of boundary strata in this work. (For more on this, see the appendix to [HZ2, (3.5)].) (0.5) We wish to remind the reader that it was necessary in [HZ2, (1.4)] to make a change of notation for the sequel, in particular this paper. In [HZ1], we used the symbol F to denote a boundary component, and associated objects such as its normalizing parabolic subgroup P_{F}, the Cayley transform c_{F}, the associated cone complex Σ_{F}, etc. carried the subscript F. Given that P = P_{F} and F are equivalent data, we decided it was more convenient to switch to writing P, c_{P}, Σ_{P}, etc. As such, one must take this change into account when tracking down notions from [HZ1]. For example, the Shimura datum denoted (G_{h,F} , X(F)) in [HZ1,1.7] is here denoted (G_{h,P} , X(P))

The notation Z_{Σ}(R) (for the closed R-stratum of the toroidal boundary) was introduced in [HZ2, (1.5.2)]. It derives from [HZ2, (1.4.11)]; there, the F -stratum should have been denoted ^{<}Z_{F, Σ} (as in 1.5.2 of [HZ1]), not ^{0}Z_{F, Σ}. Fortunately, this causes no trouble in the sequel. (0.6) There is a notational inconsistency in [HZ1] (cf. (0.5)). Along with the \overline{P} -stratum we defined the F -stratum of _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}} in [HZ1, 1.7], where F denotes the rational boundary component fixed by \overline{P}, also denoted P_{F}. However, in §5.3 of [HZ1] the F -stratum was denoted the \overline{P} -stratum. In the latter setting we had also defined R-strata for general parabolic subgroups R of G (i.e., not only for maximal parabolics) by the formula

(0.6.1)  _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}^{R} = ⋂_{\begin{aligned}& \overline{P} ⊃R \\ \overline{P} & maximal \end{aligned}}_{K_{f}^{1}}Sh(G, W \X)\overline{P} ; Σ_{1}.

Here the final \overline{P} should have been an F. However, we will retain the notation (0.6.1) and rename the F -stratum the \overline{P} (∗)-stratum, denoted _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}^{\overline{P} (∗)}. We recall that this is the union of the G(A_{f})-translates of the P_{α}-strata for all maximal parabolics P_{α} conjugate to \overline{P}. Similarly, if R is any standard parabolic subgroup of G, we let

(0.6.2)  _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}^{R(∗)} = ⋂_{\begin{aligned}& \overline{P} ⊃R \\ \overline{P} & maximal \end{aligned}}_{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}^{\overline{P} (∗)}.

In what follows, we will also use the less cumbersome notation of [HZ2, (1.5.2)] for closed boundary strata. Thus the Zariski closure of the R-stratum will be denoted Z_{Σ}(R) and the Zariski closure of the P (∗)-stratum will be denoted Z_{Σ}(P (∗)), where appropriate. (0.7) Let (G, X) be a (pure) Shimura datum. As in [HZ2] we let P(G) denote the set of rational ℚ-parabolic subgroups of G, P_{max}(G) ⊂ P(G) the subset of maximal proper parabolics; the analogous notation is used for other reductive groups. To any P ∈ P_{max}(G) we associate its Levi quotient L_{P} and subgroups G_{h,P} and G_{ℓ,P} of L_{P}, as in [HZ1, §1.2], where the “P ” was sometimes dropped; see also (0.5). As in [HZ1], we choose liftings of L_{P} to a subgroup of P, also denoted L_{P}. Sometimes we will restrict attention to standard parabolic subgroups, relative to a choice of minimal parabolic P_{0} and then the Levi subgroups L_{P} will all be assumed to contain L_{P_{0}}.

Assume G^{ad} is ℚ-simple. Then the set of standard maximal ℚ-parabolics of G P_{max}(G) is totally ordered, as follows. Fix a connected component X^{+} ⊂ X. For each P ∈ P_{max}(G) let F (P) denote the rational boundary component of X^{+} fixed by the identity component P (ℝ)^{0} of P (ℝ). Then P′ ≤ P if and only if F (P′) is contained in the closure of F (P).

As in [HZ2, (2.2)], for P ∈ P_{max}(G) we define an injection ε_{P} : P(G_{ℓ,P}) → P(G), with image denoted P_{P} (G), such that

P(G) − {G} = ∐_{P ∈P_{max}(G)} P_{P} (G).

If R ∈ P_{P} (G), then R is said to be subordinate to P, and we write Π(R) = P; this is the case if G_{h,P} is maximal among the G_{h,Q} contained in the Levi quotient of R, as Q runs through P_{max}(G) (this was stated differently in [HZ2]). For brevity, we write G_{h,R} = G_{h,P} if R is subordinate to P. Let R_{ℓ,P} = R ∩ G_{ℓ,P} (in [HZ2,(2.2.3)], this was called ι_{P} (R)). It is a ℚ-parabolic subgroup of G_{ℓ,P}, and every ℚ-parabolic of G_{ℓ,P} is of the form R_{ℓ,P} for some R subordinate to P. We also let G_{ℓ,R} denote the Levi subgroup L_{R} ∩ G_{ℓ,P} of R_{ℓ,P}. (0.8) A single object (sheaf, group, etc.) can be regarded as a complex with [at most] one non-zero term in degree 0. A sheaf S, or other such object, even if it “looks like it belongs in degree q,” is placed in degree q only by specifying the standard shift to the right: S[−q]. In general, S^{•}[−q] has in degree i what S^{•} has in degree i − q, whence the notation.

The cone C^{•} of a morphism of complexes ϕ : S^{•} → T^{•} has, by definition, as underlying sheaf S^{•}[1] ⊕ T^{•}. Often it is more natural to work with C^{•}[−1]; in that case, this shift will be specified.

1. Automorphic vector bundles on mixed Shimura varieties

In §4 of [HZ1], although we did not draw attention to this fact, we tacitly developed the basic theory of automorphic vector bundles on the mixed Shimura varieties, in the sense of Pink’s thesis [P], that arise as boundary strata of toroidal compactifications of pure Shimura varieties. The present discussion recapitulates the constructions of [HZ1] in more orderly fashion and in the full generality of [P]. The general setting includes the special case of Kuga’s families of abelian varieties with additional structure, as well as their toroidal compactifications. These will not be discussed in the sequel but the results proved here appear to have interesting applications to the construction of mixed motives.

Most of the proofs are simple adaptations of the constructions in [H1, H2, H4], as well as [HZ1, HZ2, 5.1]. Where this is the case, we keep the details to a minimum, referring to the analogous proofs in the earlier papers and thereby avoiding tedious repetition. We follow the strategy of [HZ1] by reducing purely geometric theorems about (mixed) Shimura varieties to local assertions, which we then prove on connected models of the form Γ\D.

(1.1) Mixed Shimura varieties. We refer to [P] for notation and precise definitions concerning mixed Shimura varieties. As in [HZ1, 1.6], a mixed Shimura variety is defined by a pair (Q, X), where Q is a connected algebraic group over ℚ with a three step-filtration by normal subgroups

(1.1.1)  {1} ⊂ W_{−2}Q ⊂ W_{−1}Q = R_{u}Q ⊂ W_{0}Q = Q

and X is a homogeneous space for Q(ℝ) · W_{−2}Q(ℂ) with a finite-to-one map

(1.1.2)  h : X → Hom(\underline{S}_{ℂ}, Q_{ℂ}).

We write h_{x} for h(x). It is assumed that W_{−2}Q is commutative, and for any arithmetic subgroup Γ ⊂ Q the quotient T_{Γ}(Q, X) = Γ ∩ W_{−2}Q(ℚ)\W_{−2}Q(ℂ) is viewed as the set of complex points of the split torus with character group Hom(Γ∩ W_{−2}Q, ℤ). The additional conditions satisfied by the pair (Q, X) are listed in [P, Definition 2.1] and will be recalled as needed. The mixed Shimura variety is denoted Sh(Q, X).

We write U_{Q} = W_{−2}Q, W_{Q} = W_{−1}Q, V_{Q} = W_{Q}/U_{Q}, and drop the subscript Q when it is understood, as it will be for most of the remainder of this section. Then U and V are commutative unipotent algebraic groups. In particular the exponential maps Lie(U) → U and Lie(V) → V are isomorphisms of vector groups. We let G = Q/W be the maximal reductive quotient of Q.

Let

α_{1} : Q → Q/U, α_{2} : Q/U → G, α = α_{2} ∘ α_{1} : Q → G

be the natural maps. Conditions (i) and (ii) of [P, Definition 2.1] are that (i) For any x ∈ X, α_{1} ∘ h_{x} is defined over ℝ; (ii) For any x ∈ X, α ∘ h_{x} ∘ w is a cocharacter of the center of G.

Pink’s remaining conditions imply that the pair (G, {α ∘ h_{x}|x ∈ X }) is a datum defining a pure Shimura variety. Let ρ : Q → GL(Λ) be a ℚ-rational representation. Assume the weight morphism is defined over ℚ (see (1.1.7), below). Then for any x ∈ X, the map ρ∘h_{x} defines a rational mixed Hodge structure on Λ [P, Proposition 1.4], and, as x varies, the family of ρ ∘ h_{x} defines a variation of mixed Hodge structures over X. In particular, the family satisfies Griffiths transversality. Taking ρ to be the adjoint representation, we obtain a family of mixed Hodge structures on Lie(Q). For x ∈ X, let Q_{x}^{0} ⊂ Q_{ℂ} denote the connected subgroup with Lie algebra F_{x}^{0}(Lie(Q)); then Q_{x}^{0} is the subgroup stabilizing the Hodge filtration F_{x}^{•}(Λ) for any representation (ρ, Λ). Let X̌ denote the quotient Q_{ℂ}/Q_{x}^{0}.

Let H_{X} = im(h) ⊂ Hom(\underline{S}_{ℂ}, Q_{ℂ}). The space H_{X} has a Q(ℝ) · U(ℂ)-equivariant complex structure determined by the following property: Suppose (ρ, Λ) is a faithful representation of Q. Let F(ρ) denote the variety of flags in Λ containing the Hodge filtration F_{y} attached to ρ ∘ y for any, and hence all, y ∈ H_{X}. Then the map

β′ : H_{X} → F(ρ); y ↦ F_{y}

is a complex analytic embedding [P, 1.18(a)] that naturally factors through the homogeneous space X̌(ℂ). Since h is a local diffeomorphism, this also determines a unique Q(ℝ) · U(ℂ)-equivariant complex structure on H. The map β = β′ ∘ h : X → X̌ is called the Borel morphism.

The reflex field E(Q, X) of the pair (Q, X) is the field of definition of the conjugacy class of h_{x} ∘ μ : 𝔾_{m} → Q, with μ as in (0.1), for any x ∈ X [P, Definition 11.1]. Just as in the case of pure Shimura varieties, the homogeneous algebraic variety X̌ is canonically a quotient of this conjugacy class, and for the same reasons has a natural rational structure over E(Q, X) (cf. [H1, §3]).

Let K_{f} ⊂ Q(A_{f}) be a compact open subgroup. Then the mixed Shimura variety K_{f} Sh(Q, X) is defined as a complex analytic variety, just as in the pure case, by

(1.1.3)  _{K_{f}} Sh(Q, X)(ℂ) = Q(ℚ)\(X × (Q(A_{f})/K_{f})).

As K_{f} varies, the _{K_{f}} Sh(Q, X)(ℂ) form a projective system, and one can define Sh(Q, X)(ℂ) to be the inverse limit of the _{K_{f}} Sh(Q, X)(ℂ). Alternatively, we can view Sh(Q, X)(ℂ) as shorthand for the projective system. In either case, we obtain an action of the group Q(A_{f}) on Sh(Q, X)(ℂ) as “Hecke correspondences.” However, we will mainly be concerned with the objects at finite level.

A morphism (Q, X) → (Q′, X′) of mixed Shimura data is a pair consisting of a group homomorphism Q → Q′, defined over ℚ, and a holomorphic map X → X′, compatible with the group actions on the two sides. To a morphism ϕ : (Q, X) → (Q′, X′) of mixed Shimura data we can associate holomorphic maps of mixed Shimura varieties: if K_{f} ⊂ Q(A_{f}) and K′_{f} ⊂ Q′(A_{f}) are such that ϕ(K_{f}) ⊂ K′_{f}, then there is a natural map

(1.1.4)  [ϕ] : _{K_{f}} Sh(Q, X)(ℂ) → _{K′_{f}} Sh(Q′, X′)(ℂ)

([P, 3.4]). As in the pure case, to a morphism of mixed Shimura data we obtain a map of reflex fields in the opposite direction: E(Q′, X′) ⊂ E(Q, X).

In particular, we can take Q′ = Q/U or Q′ = G = Q/W in the above construction, and Q → Q′ to be the natural projection. If we define X′ to be the quotient of X by U(ℂ), resp. by U(ℂ) · W (ℝ), we thus obtain morphisms of Shimura data (Q, X) → (Q′, X′). Writing U\X in place of U(ℂ)\X and W \X in place of U(ℂ) · W (ℝ)\X, we thus obtain a short sequence of morphisms of mixed Shimura varieties:

(1.1.5)  Sh(Q, X)(ℂ) ⟶^{π_{2}} Sh(Q/U, U\X)(ℂ) ⟶^{π_{1}} Sh(G, W \X)(ℂ).

We have already noted that (G, W \X) is a pure Shimura datum.

Among the main theorems of [P] are the existence of natural algebraic structures on the mixed Shimura varieties Sh(Q, X)(ℂ), the existence of smooth projective toroidal compactifications of the mixed Shimura varieties_{K_{f}} Sh(Q, X)(ℂ) (provided the level subgroup K_{f} is neat, in the sense of [P, 0.6]), and the existence of canonical models of Sh(Q, X)(ℂ) and its toroidal compactifications over the reflex field E(G, X) [P, Theorems 9.21, 11.18, and 12.4].^{4} We mention for the sake of completeness that any K_{f} contains a neat subgroup of finite index, and that if K_{f} is neat then any subgroup of finite index in K_{f} is also neat. The algebraic structures are “natural” in that they are compatible with the morphisms introduced above, including the Hecke correspondences Q(A_{f}) and the morphisms [ϕ] of (1.1.4) associated to morphisms ϕ : (Q, X) → (Q′, X′) of mixed Shimura data. The algebraization of the mixed Shimura varieties is actually accomplished by constructing ample line bundles on certain toroidal compactifications.

4Of course, Pink’s theorems presuppose the existence of canonical models in the pure case; see [M2] for references, as well as the recent articles of Moonen and Milne [Mo,M3], filling an apparent oversight in the literature.

The canonical model, denoted Sh(Q, X), is defined, as in the pure case, by functoriality with respect to morphisms of mixed Shimura data (T, Y) → (Q, X). Here T is a torus, the “symmetric space” Y is a finite set of points, and the canonical model of Sh(T, Y) is defined by the analogue of the Shimura-Taniyama reciprocity law for complex multiplication. The points contained in the image of a morphism of the form Sh(T, h) → Sh(Q, X), or the corresponding points at finite level, are called special points. Thus the canonical models constructed by Pink, like those of Shimura, are characterized by the reciprocity law at the special points. Details can be found in [P, 11.5].

(1.1.6) The (partial) toroidal compactifications of _{K_{f}} Sh(Q, X)(ℂ) are associated to combinatorial data, just as in the pure case [cf. HZ1, 1.4, 1.7]. Following the conventions introduced in [HZ1], the set of combinatorial data will be denoted Σ and will be called a K_{f}-admissible family of fans for (Q, X) (Pink denotes the set of data S and calls it a K_{f}-admissible partial cone decomposition; see [P, 6.4]). This aspect of Pink’s theory was discussed in [HZ1, 1.6 and 4.1] in connection with the mixed Shimura varieties that arise as boundary strata of pure Shimura varieties, and the general case is identical. The (partial) toroidal compactification of _{K_{f}} Sh(Q, X)(ℂ) associated to the K_{f}-admissible family of fans Σ is denoted K_{f} Sh(Q, X)_{Σ}(ℂ). We assume Σ satisfies the hypotheses of [P, Theorem 12.4]. Then by [loc. cit.], _{K_{f}} Sh(Q, X)_{Σ}(ℂ) exists as a complex algebraic variety, and descends to a scheme _{K_{f}} Sh(Q, X)_{Σ} over the reflex field E(Q, X). Under supplementary hypotheses on Σ, we may assume _{K_{f}} Sh(Q, X)_{Σ} to be quasi-projective; in this case we call Σ quasi-projective. Just as in the pure case, the set of all K_{f}-admissible families of fans is an inverse system under the relation of refinement. If Σ_{1} is a refinement of Σ_{2} then there is a natural morphism _{K_{f}} Sh(Q, X)_{Σ_{1}} →_{K_{f}} Sh(Q, X)_{Σ_{2}} [P, 6.7(b)]; moreover this morphism is projective [KKMS], and the set of quasiprojective Σ is cofinal in the set of all families of fans [P, Theorem 9.21].

If K_{f} is neat, then we may also choose Σ such that _{K_{f}} Sh(Q, X)_{Σ} is smooth and the complement ∂_{K_{f}} Sh(Q, X)_{Σ} =_{K_{f}} Sh(Q, X)_{Σ} −_{K_{f}} Sh(Q, X) is a divisor with normal crossings. In this case, _{K_{f}} Sh(Q, X)_{Σ} is said to be SNC.

(1.1.7) On the morphism h of (1.1.2). Say a pure Shimura datum (G_{1}, X_{1}) is standard if the map associated map h : X_{1}→Hom(\underline{S}_{ℂ}, Q_{ℂ}) of (1.1.2) is one-to-one. It follows from Proposition 2.11 of [P] that there is a standard Shimura datum (G_{1}, X_{1}) = (G, h(X)) and a Shimura datum (T, Y), with T a torus, such that (G, X) admits an embedding

(1.1.7.1)  (G, X) ↪ (G_{1}, X_{1}) × (T, Y).

In other words the map h associated to (G, X) is concentrated on the second (toric) factor.

Let (G, W \X) denote the pure Shimura datum attached to the mixed Shimura datum (Q, X). Consider the following hypotheses:

(1.1.7.2) For any h ∈ W \X, the weight morphism h ∘ w : 𝔾_{m,ℝ} → G_{ℝ} is defined over

ℚ.

(1.1.7.3) Let Z̄_{G} denote the maximal ℝ-split torus of the center of G that is ℚ-

anisotropic. Then Z̄_{G} is trivial. The Shimura datum (Q, X) will be called motivic if (1.1.7.2) holds. We assume this to be the case. Taking (G, X) in (1.1.7.1) to be (G, W \X), it is then easy to see that we may assume (G_{1}, X_{1}) and (T, Y) to be motivic as well. If G also satisfies (1.1.7.3), then we may assume this to be the case for G_{1} and T as well.

We wish to determine the failure of (T, Y) to be standard. Let h : Y→H_{Y} with H as in (1.1.3). First note that there is a unique Shimura datum (𝔾_{m}, Y(±)) such that H_{Y(±)} is the character z ↦ zz̄ of \underline{S}, and such that Y(±) has two connected components. For any integer M, the character t→t^{M} is a morphism ϕ_{M} : 𝔾_{m} → 𝔾_{m}, and there is a mixed Shimura datum (𝔾_{m}, Y(M, ±)) such that ϕ_{M} defines a morphism

(1.1.7.4)  (𝔾_{m}, Y(±))→(𝔾_{m}, Y(M, ±))

of Shimura data. (Actually, ϕ_{M} defines two morphisms. Y(M, ±) has two connected components, interchanged by complex conjugation on cyclotomic fields. Composing one choice of ϕ_{M} with exchange of the components of Y(M, ±) defines a second morphism of the form (1.1.7.4).) We let Y(M) denote the single character z ↦ (zz̄)^{M}. Then every (𝔾_{m}, Y) is of the form (𝔾_{m}, Y(M)) or (𝔾_{m}, Y(M, ±)) for some M.

Suppose T satisfies (1.1.7.3). Let T_{0} denote the maximal ℚ-anisotropic subtorus of T, T_{1} = T/T_{0}. There is a Shimura datum (T_{1}, Y_{1}) and a morphism (T, Y)→(T_{1}, Y_{1}) whose first factor is the canonical map; the datum (T_{1}, Y_{1}) is the quotient of (T, Y) by T_{0}, in the sense of [P, Proposition 2.9] By (1.1.7.3), the map π_{0}(T (ℝ))→π_{0}(T_{1}(ℝ)) is injective; it then follows from the proof of [P, loc. cit.] that the map Y→Y_{1} is also injective. On the other hand, T_{1} ⟶^{∼} 𝔾_{m}^{r} for some non-negative integer r.

Thus there is an embedding of Shimura data

(1.1.7.5)  (T, Y)→(T, H_{Y}) × (𝔾_{m}^{r}, Y_{1}).

By incorporating (T, H_{Y}) into the first factor on the right of (1.1.7.1), we see that, assuming G satisfies (1.1.7.3), there is an embedding of pure Shimura data

(1.1.7.6)  (G, X) ↪ (G_{1}, X_{1}) × (T, Y)

where (G_{1}, X_{1}) is standard and T is a ℚ-split torus.

Via (1.1.7.6), the general theory of pure Shimura data, in Pink’s sense, is reduced to the theory of standard Shimura data and the additional cases (𝔾_{m}, Y(M, ±)). We use this reduction in explaining the construction of automorphic vector bundles in the following section. (1.1.8) We will carry out certain calculations on connected components of the mixed Shimura varieties. Thus if Γ ⊂ Q(ℚ) is a congruence subgroup, and if X^{0} ⊂ X is a connected component, we let M_{Γ} = M_{Γ}(X^{0}) = Γ\X^{0}. The connected components of the mixed Shimura variety _{K_{f}} Sh(Q, X)(ℂ) are all of the form M_{Γ}(X^{0}) [P, 3.2]. The notation (X^{0}) will generally be dropped, since our assertions are independent of the choice of a connected component.

Similarly, a toroidal compactification of M_{Γ} will be denoted M_{Γ,Σ}. Here it is understood that Σ is a Γ-admissible family of fans for the pair Q, X^{0}, with the obvious definition. Alternatively, if M_{Γ} is realized as a connected component of some _{K_{f}} Sh(Q, X)(ℂ), then we may take Σ to be a K_{f}-admissible family of fans for (Q, X), and define M_{Γ,Σ} to be the closure of M_{Γ} in _{K_{f}} Sh(Q, X)_{Σ}(ℂ). Again, this distinction will be irrelevant in calculations involving connected components. (1.1.9) Boundary strata. For future reference, we introduce notation for the stratification of the toroidal boundary. For this we need to recall the classification of rational boundary components of Sh(Q, X). Let G^{ad} = G_{1} ×⋯×G_{r} be the decomposition into ℚ-simple factors. Let P_{i} ⊂ G_{i} be a ℚ-parabolic subgroup for every i, and let P ⊂ Q be the inverse image of P_{1} × ⋯ × P_{r}. Then [P, Definition 4.5] P is an admissible ℚ-parabolic subgroup of Q if every P_{i} is either equal to G_{i} or to a maximal proper ℚ-parabolic subgroup of G_{i}. As in the pure case (see (0.7)), we let P(Q) denote the set of admissible ℚ-parabolics of Q.

In particular, every admissible ℚ-parabolic subgroup of Q is the pullback of an admissible ℚ-parabolic subgroup of G. In what follows we refer to (0.6) for notation concerning the boundary strata in the pure case. Let \overline{P} be an admissible ℚ-parabolic subgroup of G, P the corresponding admissible parabolic of Q. We define the \overline{P} -stratum _{K_{f}^{1}}Sh(G, W \X)\overline{P} ; Σ_{1} of _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}} as in [HZ1, 1.7], and define the P -stratum of _{K_{f}} Sh(Q, X)_{Σ} to be

(1.1.10)  _{K_{f}} Sh(Q, X)_{Σ}^{P} = (π_{2,Σ} ∘ π_{1,Σ})^{−1}(_{K_{f}^{1}}Sh(G, W \X)\overline{P} ; Σ_{1}).

We note that these strata are in general not Zariski closed.

If now P is any ℚ-parabolic subgroup of Q, we define the P -stratum of_{K_{f}} Sh(Q, X)_{Σ} by analogy with (1.1.10); the P (∗)-stratum _{K_{f}} Sh(Q, X)_{Σ}^{P (∗)} of _{K_{f}} Sh(Q, X)_{Σ} is defined in the same way, with \overline{P} replaced by \overline{P} (∗).

(1.2) Construction of automorphic vector bundles. We keep the notation of §1.1. Henceforward, we make the following standard Hypothesis (1.2.1). The Shimura datum (Q, X) is motivic (1.1.7.2) and satisfies condition (1.1.7.3).

Hypothesis 1.2.1 guarantees that any algebraic representation of G gives rise to a rational local system over Sh(Q, X)(ℂ), in the sense of [H1, 3.14]. The hypothesis also allows us to define the standard principal bundle I(Q, X)(ℂ) over Sh(Q, X)(ℂ) as the complex analytic variety

(1.2.2)  I(Q, X)(ℂ) = \underleftarrow{lim} Q(ℚ)\(Q(ℂ) × X × (Q(A_{f})/K_{f})).

Here as in (1.1.3), the group Q(ℚ) acts diagonally on the left on the product of the three factors. That this is indeed a principal bundle follows in the usual way from Hypothesis 1.2.1 (cf. [M2, Proposition 3.3]). Again, this can be viewed as an inverse system of principal Q(ℂ)-bundles

K_{f} I(Q, X)(ℂ) → _{K_{f}} Sh(Q, X)(ℂ).

Remark. One obtains slightly more generality by dropping hypothesis (1.2.1) and restrict attention to algebraic representations of G trivial on the subgroup Z̄(G) defined in (1.1.7.3). We leave it to the interested reader to carry out the necessary modifications.

The natural projection I(Q, X)(ℂ) → Sh(Q, X)(ℂ) is denoted q = q_{Q,X}. On the other hand, there is a natural map p = p_{Q,X} : I(Q, X)(ℂ) → X̌(ℂ) defined in terms of (1.2.2) by

(1.2.3)  p(q, x, q_{f}) = q^{−1} · β(x), q ∈ Q(ℂ), x ∈ X , q_{f} ∈ (Q(A_{f})/K_{f}).

The right-hand side is obviously invariant under left-translation by Q(ℚ) and defines a holomorphic map.

Proposition (1.2.4). The standard principal bundle I(Q, X)(ℂ) can be given a natural structure of algebraic principal bundle I(Q, X)_{ℂ} over Sh(Q, X)_{ℂ}. Moreover, I(Q, X)_{ℂ} has a canonical Q(A_{f})-equivariant model I(Q, X) over the reflex field E(Q, X), and the morphisms q_{Q,X} and q_{Q,X} descend to yield a canonical diagram

I(Q, X)

p ↙  ↘^{q}
X̌  Sh(Q, X)

over E(Q, X).

(1.2.5) Explanation-Definition. The canonical model to which we refer in the proposition is defined by combining the definitions of Pink (for canonical models of mixed Shimura varieties) and Milne (for canonical models of standard principal bundles over pure Shimura varieties, see [M1]). Unfortunately, a complete explanation requires additional notation. We take as given the existence of a canonical model on I(T, Y) when T is a torus. When Y is a point, this canonical model is defined as in [M1] in terms of the period torsor. The discussion in (1.1.7) reduces the general case to the case where T = 𝔾_{m}, which will be treated in the appendix (1.2.A), below.

If (Q′, X′) → (Q, X) is an embedding of mixed Shimura data then E(Q, X) ⊂ E(Q′, X′) [P, 11.2(b)]. Then a canonical model on I(Q, X) is a Q(A_{f})-equivariant model such that, for any embedding of mixed Shimura data (T, Y) → (Q, X), with T a torus, the natural morphism I(T, Y) → I(Q, X) descends to E(T, Y). One verifies as for pure Shimura varieties that the canonical model is unique, if it exists.

Proof of Proposition 1.2.4. For any positive integer g, we let (P_{2g}, X_{2g}) be the mixed Shimura datum defined in [P, 2.25]. It is a mixed Shimura datum associated to the genus g (maximal) boundary component of the Siegel modular Shimura variety of genus g + 1, denoted Sh(GSp(g + 1), 𝔖_{g+1}^{±}). Here GSp(g + 1) is the group of similitudes of a 2(g + 1)-dimensional symplectic space and 𝔖_{g+1}^{±} is the union of the Siegel upper and lower half-spaces of genus g + 1. We can realize the parabolic subgroup P_{2g} ⊂ GSp(g+1) as the stabilizer of a rational genus g boundary component of 𝔖_{g+1}^{±}. For a cofinal subset of the set of level subgroups K_{f} ⊂ P_{2g}(A_{f}), the associated mixed Shimura variety _{K_{f}} Sh(P_{2g}, X_{2g}) can thus be realized as a union of boundary strata of a toroidal compactification of Sh(GSp(g + 1), 𝔖_{g+1}^{±}) (at an appropriate finite level, depending on K_{f}). Since the center of the unipotent radical of P_{2g} is one-dimensional, the realization as a (union of) boundary strata does not depend on the choice of toroidal compactifications.

Now when (Q, X) is a pure Shimura datum and the natural map X → H is the identity – i.e., (Q, X) is a standard Shimura datum – then the proposition is proved in [H1] and [M1]. When Q is a torus, the proposition is tautologically true, whether or not X is connected. On the other hand, when _{K_{f}} Sh(Q, X) is a (union of) boundary strata of a toroidal compactification attached to a rational boundary component of a pure Shimura variety, then the canonical model of I(Q, X)_{ℂ} is constructed in [HZ1, Lemma 4.5.8]. The fact that this model is canonical is an immediate consequence of [HZ1, Lemma 4.6.9], since, when (T, h) is as in 1.2.5, the morphism Sh(T, h) → Sh(Q, X) factors through some section of the natural map π_{1} ∘ π_{2} : Sh(Q, X) → Sh(G, W \X).

We remark that the Q(A_{f})-equivariance is not stated explicitly in [HZ1], and indeed the statement needs to be modified for boundary strata of toroidal compactifications to take account of the combinatorial data defining the compactification. We will only apply [HZ1, Lemma 4.5.8] to the case of (P_{2g}, X_{2g}). The fan in this case is uniquely determined. Therefore, the P_{2g}(A_{f})-equivariance follows immediately from the GSp(g +1)(A_{f})-equivariance of the canonical model of the standard principal bundle over Sh(GSp(g + 1), 𝔖_{g+1}^{±}).

It follows in the usual way that if (Q′, X′) ↪ (Q, X) is an embedding of Shimura data, then the existence of a canonical model for I(Q, X) implies the existence of a canonical model for I(Q′, X′). It thus follows from what has been established up to this point that, if (Q, X) admits an embedding of the form

(1.2.6)  (Q, X) → (G_{1}, X_{1}) × (T, Y) × ∏_{i=1}^{n}(P_{2g}, X_{2g})

where (G_{1}, X_{1}) is a standard Shimura datum, T is a ℚ-split torus, and n is a non-negative integer, then the proposition holds for (Q, X).

Lemma 2.26 of [P] asserts that, we can replace (Q, X) by a new pair (Q′, X′), where Q′ is an extension of Q by U_{0} ≃ 𝔾_{a} (contained in the center of the unipotent radical of Q′) of a specific type, called a (P_{0}, X_{0}) → (𝔾_{m,ℚ}, H_{0})-torsor, such that (Q′, X′) admits an embedding of the form (1.2.6); the fact that we can take T ℚ-split is (1.1.7.6). Thus we have the Q′-torsor I(Q′, X′) over Sh(Q′, X′) with its canonical model over E(Q′, X′), which is easily checked to equal E(Q, X). The quotient of I(Q′, X′) by U_{0} is then a Q-torsor over Sh(Q′, X′), which is in turn a 𝔾_{m}-torsor over Sh(Q, X). Over ℂ, U_{0}\I(Q′, X′) is obviously isomorphic to the pullback of I(Q, X)_{ℂ} from Sh(Q, X). Thus the 𝔾_{m}-action on Sh(Q′, X′) lifts canonically to an action on U_{0}\I(Q′, X′) and thus defines an E(Q, X)-rational model I(Q, X) of I(Q, X)_{ℂ}. Since any morphism (T, Y) → (Q, X), with T a torus, lifts to a morphism (T, Y) → (Q′, X′), the canonicity of I(Q, X) follows from that of I(Q′, X′).

It remains to show that the morphism p : I(Q, X) → X̌ can be descended to E(Q, X). This follows again by reduction to the individual factors on the righthand side of (1.2.6). For the first factor this is in [H1] and [M1], for the second factor it’s trivial, and for the third factor it is contained in Lemma 4.5.8 of [HZ1]. This completes the proof.

Given the diagram in Proposition 1.2.4, automorphic vector bundles can be constructed on Sh(Q, X) by the procedure described in [H1] and [M1], which we now recall. Let W be a Q-homogeneous vector bundle on X̌. Then p^{∗}W is a Q-equivariant vector bundle on I(Q, X), hence descends to a Q(A_{f})-equivariant vector bundle [W] on Sh(Q, X). The functor W ↦ [W] is rational over E(Q, X), by construction, and defines canonical models, just as in [M1], on the complex points of [W], which just as in the classical case can be identified with a locally homogeneous vector bundle:

(1.2.7)  [W](ℂ)⟶^{∼}(Q(ℚ)\β^{∗}(W) × (Q(A_{f})/K_{f})).

A Q-homogeneous vector bundle on X̌ is called locally-finite if it is the direct limit of finite-rank homogeneous vector bundles on X̌. The functor above obviously is defined on locally-finite homogeneous vector bundles and satisfies [\underrightarrow{lim} W_{α}] = \underrightarrow{lim}[W_{α}]. Defining [W](ℂ) by (1.2.7), we resume this discussion as follows: Proposition (1.2.8). The functor W ↦ [W](ℂ), defined by (1.2.7), maps the tensor category of locally finite Q-homogeneous algebraic vector bundles on X̌ to the tensor category of Q(A_{f})-equivariant algebraic vector bundles on Sh(Q, X). This functor is rational over E(Q, X) and is functorial with respect to morphisms (Q′, X′) → (Q, X) of mixed Shimura data. When (Q, X) is a pure Shimura datum, the functor coincides with the one defined in [H1, M1]. In particular, the bundles [W] have canonical models in the sense of [M1].

The final assertion in (1.2.8) is an immediate consequence of the others, since the canonical models are defined with respect to embeddings of pure Shimura data of the form (T, h), with T a torus. Bundles of the form [W] are called automorphic vector bundles. The automorphic vector bundles form a tensor subcategory of the category of all Q(A_{f})-equivariant vector bundles on Sh(Q, X).

Let (ρ, L) be an algebraic representation of Q. We say (ρ, L) is locally finite if it is the direct limit of finite-dimensional (algebraic) representations of Q. An automorphic vector bundle [W] on Sh is called flat if W is isomorphic to a Qhomogeneous vector bundle on X̌ of the form L ⊗ OX̌, where (ρ, L) is a locallyfinite representation of Q. We then write [W] = L̃. Equivalently, on any connected component M_{Γ} ⊂ Sh(ℂ), [W] is the direct limit of vector bundles associated to finite-dimensional algebraic representations of the fundamental group Γ of M_{Γ} (here “algebraic representation of Γ” is shorthand for “restriction to Γ of an algebraic representation of Q.”) A flat vector bundle always has a canonical flat connection ▽ = ▽_{L} : L̃ → L̃ ⊗ Ω_{Sh}^{1}, induced by functoriality from the constant connection

1 ⊗ d : L ⊗ OX̌ → L ⊗ Ω1 ; X̌.

The standard principal bundle I(Q, X) has the following functorial description. Let (ρ, L) be a faithful ℚ-rational representation with contragredient (ρ^{∗}, L^{∗}). Let s_{i} ∈ L^{⊗n_{i}} ⊗ (L^{∗})^{⊗m_{i}} (i = 1, …, r) be a finite set of ℚ-rational tensors such that Q is the subgroup of GL(L) stabilizing all s_{i}. Let 1 denote ℚ, viewed as a trivial representation of Q. For each i, the morphism of Q-representations

α_{i} : 1 → L^{⊗n_{i}} ⊗ (L^{∗})^{⊗m_{i}}; α_{i}(1) = s_{i}

defines by functoriality an injective homomorphism

(1.2.9)  α̃_{i} : O_{Sh} → L̃^{⊗n_{i}} ⊗ (L̃^{∗})^{⊗m_{i}}

of automorphic vector bundles. Let s̃_{i} = α̃_{i}(1), i = 1, …, r. It is then tautological that I(Q, X) represents the functor that, to any Sh-scheme T, associates

(1.2.10)  {f : L̃_{T} ⟶^{∼}L_{T} | (f^{⊗n_{i}} ⊗ (f^{∗})^{⊗m_{i}})(s̃_{i}) = s_{i} ⊗ 1}.

Here L_{T} is the constant bundle L ⊗ O_{T} and the remaining notation is obvious. In fact, the construction of the canonical model of I(Q, X) in [H1,M1] in the pure case proceeds by constructing the flat filtered automorphic vector bundle L̃ and showing that it has an appropriate kind of canonical model; in particular, that the [s_{i}] are rational over E(Q, X). Then I(Q, X) is defined by (1.2.10), and is shown as in [H1, Remark 4.9(1); M2, Remark 3.3] to be independent of the choice of (ρ, L).

The morphism π_{2} : Sh(Q, X) → Sh(Q/U, U\X) is a locally constant pro-torus fibration over Sh(Q/U, U\X) with fiber T = \underleftarrow{lim} U(ℚ)\U(ℂ) × U(A_{f})/K_{U,f}. Here K_{U,f} runs over the set of open compact subgroups of U(A_{f}). At finite level K_{f}, π_{2} is a locally constant torus fibration. It follows just as in [HZ1, (3.2.1)] that

(1.2.11)  [W] = π_{2}^{∗}(π_{2,∗}([W])^{T}).

Here T acts naturally on the direct image π_{2,∗}([W]) (cf. [SGA 3]) and the superscript T designates the subsheaf of T -invariant sections.

This can be globalized, following [HZ1, §4.6]. The subgroup W ⊂ Q acts freely on the principal bundle I(Q, X). We let I_{2}(Q, X) (resp. I_{1}(Q, X)) denote the quotient of I(Q, X) by U (resp. by W). Then I_{2}(Q, X) (resp. I_{1}(Q, X)) represents the functor defined by analogy with (1.2.10), in which the faithful representation (ρ, L) of Q is replaced by a representation of Q that factors through a faithful representation of Q/U (resp. Q/W). It follows immediately that

(1.2.12) I_{2}(Q, X)⟶^{∼}π_{2}^{∗}(I(Q/U, U\X)), I_{1}(Q, X)⟶^{∼}(π_{1} ∘ π_{2})^{∗}(I(Q/W, W \X)),

the isomorphisms being canonical and, in particular, defined over E(Q, X). On the other hand, there are canonical isomorphisms

(1.2.13)  U\X̌⟶^{∼}(U\X)^{∨}, W \X̌⟶^{∼}(W \X)^{∨}.

The isomorphisms (1.2.13) can be defined over ℂ tautologically, in terms of the definition of X̌ as a homogeneous space under Q(ℂ); it is then easy to see that the isomorphisms descend to E(Q, X). Now any Q-equivariant vector bundle W on X̌ is necessarily U-equivariant. Since U acts freely on X̌, the natural map π̌_{2} : X̌ → (U\X)^{∨} makes X̌ a U-torsor over (U\X)^{∨}. It follows that

W = π̌_{2}^{∗}(W_{U}),

for some Q/U-homogeneous vector bundle W_{U} on (U\X)^{∨}. It is then easy to see that, in the notation of (1.2.11),

(1.2.14)  (π_{2,∗}([W])^{T} ⟶^{∼}[W_{U}].

In particular, π_{2,∗}([W])^{T} is an automorphic vector bundle over Sh(Q/U, U\X). The analogue for π_{1} is more complicated and will be discussed in §1.4.

(1.2.15) Remark The generalization of Langlands’ conjecture on conjugation of Shimura varieties to the setting of mixed Shimura varieties has been carried out in [W1, II, 5]. Wildeshaus also constructs flat automorphic vector bundles on mixed Shimura varieties [W1, II,6] and determines their behavior with respect to the action of Aut(ℂ).

(1.2.16) Let x ∈ X and let Q_{x}^{0} ⊂ Q be its stabilizer, as in §1. In the diagram (1.2.7), the homogeneous vector bundle β^{∗}(W) is determined by the isotropy representation of Q_{x}^{0} on the fiber W_{x}. We say W is fully decomposed if the isotropy representation at some point x is trivial on the unipotent radical R_{u}Q_{x}^{0} of Q_{x}^{0}. This condition is evidently independent of the choice of x, and we then say that the automorphic vector bundle [W] is fully decomposed.

We will need a more general condition in practice. As above, we write W = W_{−1}Q, U = W_{−2}Q.

Definition (1.2.16.1). We say W satisfies a Kostant condition at x if there are i) a reductive group K_{x}, ii) a parabolic subgroup Π_{x} ⊂ K_{x}, iii) an injective map α_{x} : Q_{x}^{0} ∩ W →Π_{x} with α(Lie(Q_{x}^{0} ∩ W)) = R_{u} Lie(Π_{x}), iv) a finite-dimensional representation σ_{x} of K_{x} on W_{x}

such that the isotropy representation of Lie(Q_{x}^{0} ∩ W) on W_{x} is equivalent to σ_{x} |_{Lie(Π_{x})} ∘α_{x}.

Again, this condition is independent of the choice of x, so we simply say that the automorphic vector bundle [W] satisfies a Kostant condition. Such a condition serves to provide the splitting of spectral sequences at E_{2}, as in Corollary 1.4.5 below. This splitting is canonical, given x and items (1.2.16.1,(i)–(iv)) (x determines a canonical Levi subgroup of Q). Indeed, the derived functor RHom_{Lie(Q_{x}^{0}∩W)}(ℂ, W_{x}) splits canonically as the sum of its cohomology objects, for one then gets an embedding

(1.2.16.2)  H^{•}(Lie(Q_{x}^{0} ∩ W), W_{x}) ↪ C^{•}(Lie(Q_{x}^{0} ∩ W), W_{x}),

where C^{•} is the standard Lie algebra cochain complex and H^{•} has the zero differential, inducing the identity on cohomology. Remark (1.2.16.3) The consequence (1.2.16.2) of the Kostant condition is really just [K,(5.7.4)]. In its original form, it was tacitly invoked several times in the calculation of boundary cohomology of the pure Shimura variety Sh(G, X) in [HZ1,§3]. (To be specific, it enters there in at least 3.5.12, 3.7.8 via 3.7.5, 3.13.4.5, and 3.14.2—though not 3.5.8.) The mixed Shimura varieties there were of the form (P′_{F} , X(D_{F})) attached to a rational boundary component F of X, with P′_{F} = G_{h,F} W_{F} and X(D_{F}) an enlarged Siegel domain of the third kind containing X (see [HZ1, 1.2.3] for this notation). We write W = W_{F}, U = U_{F}. If x ∈ X is viewed as a point in X(D_{F}), then the natural map

Q_{x}^{0} ∩ W → W/U = gr_{−1}W

is injective and identifies Lie(Q_{x}^{0} ∩ W) with the anti-holomorphic tangent space 𝔳_{x}^{−} [HZ1,1.8.4] of the abelian variety corresponding to the point x. In this case, the group K_{x} is the complexification of the connected stabilizer of x ∈ X in G(ℝ); i.e., of the maximal compact (mod center) subgroup of G(ℝ), the parabolic subgroup Π_{x} is the group denoted Q_{F,x} in [HZ1,1.8.7], and Lie(R_{u}Q_{F,x}) = 𝔰_{x}; and the map α_{x} : 𝔳_{x}^{−}→Lie(R_{u}Q_{F,x}) is defined via the Cayley transform in [HZ1,3.5]. In [HZ1: 3.5.12], the embedding (1.2.16.2) is tacitly used, though it might have been more helpful to quote [K: 5.7.4] explicitly. (1.2.16.4) Variant. Let S_{x} ⊂ Q_{x}^{0} ∩ W be a subgroup with 𝔰_{x} = Lie(S_{x}). Suppose that the analogues of (1.2.16.1, (i)–(iv)) hold, with Q_{x}^{0} ∩ W replaced by S_{x} and Lie(Q_{x}^{0} ∩ W) replaced by 𝔰_{x}. We then say [W] satisfies a Kostant condition relative to S_{x}, (or 𝔰_{x}). The analogue of (1.2.16.2) then holds for 𝔰_{x}: there is a canonical splitting

(1.2.16.5)  H^{•}(𝔰_{x}, W_{x}) ↪ C^{•}(𝔰_{x}, W_{x}).

A condition of this kind occurs in the proof of Proposition 1.5.24. (1.2.A) Appendix: Periods and rationality. The articles [H1] and [M1] define canonical rational structures on automorphic vector bundles in terms of CM motives. This construction is adapted in section 4.3 of [HZ1] to the mixed Shimura varieties treated there. We briefly show how this construction extends to general mixed Shimura varieties.

In the previous version of the present article, the construction of standard principal bundles on mixed Shimura varieties was reduced to the case of pure Shimura varities, by the procedure outlined in the proof of Proposition 1.2.4. We are grateful to the referee for pointing out that the literature only considers pure Shimura varieties in the standard definition, and not the more general definition due to Pink. The purpose of the present appendix is to fill this gap in the literature with an exposition of Pink’s constructions. As explained in (1.1.7), it suffices to treat the case of a pure Shimura datum (T, Y), where T is a ℚ-split torus. However, it is no more difficult to treat a general torus, and we do so now.

If (Q, X) is a general mixed Shimura datum, and if (x, q_{f}) ∈ X × Q(A_{f}), we let [x, q_{f}] denote the corresponding point in Sh(Q, X)(ℂ). We begin by recalling the period formalism, as described in [HZ1, §4.3]. First suppose (T, h) is a Shimura datum with T a torus and h a point. We write ȟ for the “compact dual”, as above. Let W be a T -homogeneous vector bundle on ȟ, and let [W] be the corresponding automorphic vector bundle. The homogeneous vector bundle W defines a representation σ = σ_{W} of T and a period element

p(h, σ) = p(h, σ_{W}) ∈ Aut([W]|_{Sh(T,h)}(ℂ)),

well-defined modulo σ_{W}(T (ℚ)), that compares the de Rham and Betti rational structures of [W]|_{Sh(T,h)}(ℂ). In the notation of [HZ1, (4.3.6)], we have

(1.2.A.1)  H_{DR}(𝔐(σ)/Sh(T, h))_{[h,q_{f}]} = p(h, σ) · H_{B}(𝔐(σ)/Sh(T, h))_{[h,q_{f}]}.

Here q_{f} is a variable point in T (A_{f}), as above.

We note that either side of (1.2.A.1) can be taken to be defined independently. Over ℂ, we just have

(1.2.A.2)  H_{B}(𝔐(σ)/Sh(T, h))(ℂ) = Γ(Sh(T, h)(ℂ), W(ℂ)),

with W(ℂ) defined by the formula (1.2.7). There is a reflex field E(W) ⊃ E(T, h), such that both sides of (1.2.A.1) have natural E(W)-rational structures. (In practice, we can take E(W) to be the composite of E(T, h) with a field of definition of the algebraic representation σ.) Then H_{B}(𝔐(σ)/Sh(T, h)) can be defined, in terms of (1.2.A.2), by

(1.2.A.3)  H_{B}(𝔐(σ)/Sh(T, h)) = Γ(Sh(T, h)(ℂ), W(E(W))),

where by W(E(W)) we mean the local system in E(W)-vector spaces

\underleftarrow{lim}_{K_{f}} T (ℚ)\W(E(W)) × (T (A_{f})/K_{f})).

On the other hand, the left-hand side of (1.2.A.1) is given (a posteriori) by

(1.2.A.4)  H_{DR}(𝔐(σ)/Sh(T, h)) = Γ(Sh(T, h), [W]),

where here the relevant structures are the E(T, h)-scheme Sh(T, h) and the (rational) vector bundle [W]. The content of (1.2.A.1) is that there exists a theory of period invariants p(h, σ) that behaves naturally with respect to morphisms of Shimura data and tensor products. These invariants, initially introduced (up to algebraic factors) by Shimura, are provided by the theory of abelian varieties with complex multiplication, and are described in detail in [H1].

Now let (T, Y) be a general Shimura datum, in Pink’s sense, with T a torus; as always we assume hypothesis (1.2.1). We write h for the singleton H_{Y}; thus there is a morphism of Shimura data (T, Y)→(T, h), inducing a morphism of Shimura varieties η : Sh(T, Y)→Sh(T, h) as well as a T -equivariant identification of compact duals Y̌→ȟ. If W is a T -homogeneous vector bundle on Y̌ = ȟ, then we define the corresponding automorphic vector bundle [W]_{Y} on Sh(T, Y) by

(1.2.A.5)  [W]_{Y} = η^{∗}[W].

Here [W] on the right-hand side is the automorphic vector bundle on Sh(T, h). This is the only definition compatible with the functoriality assertion of Proposition 1.2.8. The element p(h, σ) of (1.2.A.1) lifts to an automorphism of [W]_{Y} which we denote p(h, σ)_{Y}. We then tautologically obtain

(1.2.A.6) H_{DR}(𝔐(σ)/Sh(T, Y))_{[y,q_{f}]} = p(h, σ_{Y}) · H_{B}(𝔐(σ)/Sh(T, Y))_{[y,q_{f}]}.

for y ∈ Y, where both sides of (1.2.A.6) are defined in terms of the corresponding sides of (1.2.A.1) by pullback.

This apparently simplistic definition of the period invariant has a simple justification. There is an isomorphism

(1.2.A.7)  Γ(Sh(T, Y), [W]_{Y})⟶^{∼}O_{Sh(T,Y)} ⊗_{O(Sh(T,h))} Γ(Sh(T, h), [W]).

The period term is concentrated in the second factor on the right; the first factor accounts for the reciprocity law of Sh(T, Y), as defined by Pink.

Now we consider the period invariant in general. Let (Q, X) be a mixed Shimura datum. Let W be a Q-homogeneous vector bundle on X̌, and let [W] be the corresponding automorphic vector bundle. As in the pure case, there is a canonical “periods” isomorphism

(1.2.A.8)  Per_{(x,q_{f})} : W_{β(x)} ⟶^{∼} [W]_{[x,q_{f}]}

(cf. [HZ1, (4.3.5)]). Suppose W is rational over the extension E(W) of E(Q, X), as Q-homogeneous vector bundle. By Proposition 1.2.8, [W] has a canonical model over E(W). This model has the following description in terms of the period elements: For any map (T, Y) ↪ (Q, X), with T a torus, let E(T, Y; W) = E(T, Y) · E(W). Then for any field L ⊃ E(T, Y; W), x ∈ Y, and any t_{f} ∈ T (A_{f}), we have

(1.2.A.9)  [W](L)|_{[x,t_{f}]} = p(h, σ_{W}) · Per_{(x,t_{f})}(W_{β(x)}(L)) ⊂ [W]|_{[x,t_{f}]}(ℂ).

This both summarizes and generalizes the discussion that follows [HZ1, (4.3.6)].

(1.3) Canonical extension of automorphic vector bundles. Let _{K_{f}} Sh(Q, X) ⊂ K_{f} Sh(Q, X)_{Σ} be a (partial) toroidal compactification as an algebraic variety, as in (1.1.6). We retain Hypothesis (1.2.1) and assume for the remainder of §1 that K_{f} is neat, in the sense of [P, 0.6], and that _{K_{f}} Sh(Q, X)_{Σ} is an SNC compactification. We write Sh = _{K_{f}} Sh(Q, X), Sh_{Σ} = _{K_{f}} Sh(Q, X)_{Σ}, ∂Sh_{Σ} = ∂_{K_{f}} Sh(Q, X)_{Σ}.

After (1.3.5), and for the remainder of the paper, we also assume _{K_{f}} Sh(Q, X)_{Σ} to be compact, unless an explicit statement is made to the contrary.

Our hypothesis that K_{f} is neat implies, as in the pure case (cf. the proof of [H2, Theorem 4.2]) that the connection ▽_{L} has unipotent monodromy. Deligne’s existence theorem [De1] thus provides a unique extension of L̃ to a vector bundle L̃_{Σ} over Sh_{Σ} such that ▽ extends to a connection ▽_{Σ} with regular singularities and nilpotent connection forms along ∂Sh_{Σ}.

Definition (1.3.1). Let [W] be an automorphic vector bundle on Sh. An admissible metric on [W] is a hermitian metric whose pullback, under the natural map X × Q(A_{f}) → Sh, is a Q(ℝ) · U(ℂ) × Q(A_{f})-invariant hermitian metric on β^{∗}(W × X × Q(A_{f})).

The stabilizer Stab(x) in Q(ℝ) · U(ℂ) of any point x ∈ X is a compact Lie group (Pink doesn’t state this explicitly, but it is clear from Lemma 1.17 of [P]; cf. the proof of [P, Prop. 2.11].). Therefore, admissible metrics exist and their restrictions to indecomposable C^{∞} summands of [W] are unique up to constant multiples. Note that the automorphic vector bundles that are indecomposable as C^{∞} automorphic bundles are parametrized by irreducible representations of the compact group Stab(x), for one (or any) x ∈ X. The indecomposable holomorphic automorphic bundles are parametrized by locally-finite indecomposable representations of the algebraic subgroup Q_{x}^{0} ⊂ Q, defined in (1.1).

Lemma (1.3.2). (i) Let [W] be an automorphic vector bundle on Sh. Then [W] has an admissible metric. (ii) Let h(·, ·) be an admissible metric on the flat vector bundle L̃. Let B ⊂ Sh_{Σ} be an open polydisk with coordinates z_{1}, … z_{n} chosen so that B ∩ ∂Sh_{Σ} is the union of the coordinate hyperplanes defined by z_{1}, …, z_{r} for some r ≤ n. Then

(1.3.3) Γ(B, L̃_{Σ}) = {s ∈ Γ(B∩Sh, L̃)||h(s, s)| = O(P (|log(z_{1})|, …, |log(z_{r})|)) for some polynomial P }.

Proof. Part (i) has already been explained. To prove Part (ii), we reduce to the case of an equivariant torus embedding. Indeed, let Q′ ⊂ Q be a rational boundary subgroup and let X′ be the corresponding rational boundary component of X, in the sense of [P, §4]. We let U′ ⊂ W′ = R_{u}Q′ ⊂ Q′ be the canonical filtration of Q′, and let π′_{2} : Sh(Q′, X′) → Sh(Q′/U′, U′\X′) be the canonical map, representing Sh(Q′, X′) as the locally constant pro-torus fibration over Sh(Q′/U′, U′\X′) with structure group T′, as in the discussion preceding (1.2.11). There is a natural embedding X̌′ ⊂ X̌ (cf. [HZ1, Lemma 4.5.8]), extending the inclusion of β(X′) in X̌ as a Siegel domain of the third kind. There is also a neighborhood Sh(Q′, X′)_{Σ_{Q′}}^{+} of the boundary ∂Sh(Q′, X′)_{Σ_{Q′}} admitting a natural map

(1.3.4)  f_{Q′} : Sh(Q′, X′)_{Σ_{Q′}}^{+} → Sh_{Σ}.

The map f_{Q′} is a local isomorphism in a neighborhood of the part of the boundary of Sh_{Σ} corresponding to Q′.

We assume Q′ chosen so that the polydisk B lies in the image of f_{Q′}. The flat vector bundle L̃ is defined by the representation ρ, say of Q. The restriction of ρ to Q′ defines a flat vector bundle (L̃)′ on Sh(Q′, X′) whose restriction to Sh(Q′, X′)_{Σ_{Q′}}^{+} ∩ Sh(Q′, X′) is equivalent to f_{Q′}^{∗}(L̃). Then it is clear by uniqueness that the admissible metric h(·, ·) on L̃ pulls back to the restriction to Sh(Q′, X′)_{Σ_{Q′}}^{+} of an admissible metric on (L̃)′. Thus we may assume Q′ = Q and that B is not contained in the image of f_{Q′′} for any proper boundary subgroup Q′′ of Q. In other words, possibly shrinking B, the map π_{2} can be written as a product

π_{2} : B⟶^{∼}B_{0} × T_{Σ(T )}

where B_{0} is open in Sh(Q/U, U\X) and T ↪ T_{Σ(T )} is an equivariant torus embedding, with T a torus of the form U(ℚ)\U(ℂ) × U(A_{f})/K_{U,f} as in the discussion at the end of §1.2. Near B the local system is thus defined by a representation of the fundamental group of T . In this way, we may assume Q = U, Sh is a torus of the form U(ℂ)/Λ, for some lattice Λ in U(ℝ), L̃ is the local system defined by a representation (ρ, L) of Λ, and h(·, ·) is a U(ℂ)-invariant metric. The representation ρ is the restriction to Λ of a unipotent representation of the algebraic group U. Moreover, we may assume n = r and Sh ↪ Sh_{Σ} is isomorphic to the obvious embedding ℂ^{×,n} → ℂ^{n}, with coordinates z_{1}, …, z_{n}. In particular, Sh_{Σ} is affine, hence L̃_{Σ} generated by its global sections.

Let j : U(ℂ) × U(A_{f}) → Sh denote the natural map. We fix a basepoint, say o, in U(ℂ), and a basis {e_{1}, …, e_{N}} of L, such that h(e_{i}, e_{j})_{o} is the identity matrix. It follows from the U(ℂ)-invariance of h(·, ·) that, with respect to the trivialization j^{∗}(L̃)⟶^{∼}U(ℂ) × L, the matrix of h(e_{i}, e_{j})_{u} is polynomial in the variable u ∈ U(ℂ), hence have logarithmic growth in terms of the coordinates z_{1}, …, z_{n}. Now let {v_{1}, …, v_{N}} denote a basis of global sections of L̃_{Σ}, and express the v_{i} in terms of the horizontal sections e_{j} over the universal cover U(ℂ) of Sh:

v_{i} = ∑_{j} f_{ij}e_{j},

where f_{ij} are functions on U(ℂ). Since the monodromy of L̃ is unipotent, the f_{ij} have logarithmic growth in terms of z_{1}, …, z_{n}. The inclusion of the left-hand side of (1.3.3) in the right-hand side is now clear. The opposite inclusion then follows because any section satisfying an inequality of the indicated form has removable singularities with respect to the basis {v_{1}, …, v_{N}}.

A local section of L̃ satisfying an inequality of type (1.3.3) is said to be slowly increasing.

In (ii) of the following Definition-Proposition, we let (Q′, X′), U′, and so on be as in the proof of Lemma 1.3.2.

Definition-Proposition (1.3.5). Let [W] be an automorphic vector bundle on Sh. A canonical extension [W]_{Σ} of [W] over Sh_{Σ} is a vector bundle over Sh_{Σ} satisfying one of the following equivalent conditions:

(i) If h(·, ·) be an admissible metric on [W] and if B is as in Lemma (1.3.2, ii), then

Γ(B, [W]_{Σ}) = {s ∈ Γ(B∩Sh, [W]) : |h(s, s)| = O(P (|log(z_{1})|, … |log(z_{r})|)) for some polynomial P };

the polynomial P above may depend on s.

(ii) Let (Q′, X′) be a rational boundary pair for (Q, X), as above. Write Σ = ⋃ Σ_{P}, the union taken over the rational boundary subgroups P of Q. If σ ∈ Σ_{Q′}, let Sh(Q′, X′)_{σ} ⊂ Sh(Q′, X′)_{Σ_{Q′}} denote the corresponding partial compactification, and let

π′_{2,σ} : Sh(Q′, X′)_{σ} → Sh(Q′/U′, U′\X′)

denote the natural extension of π′_{2} (cf. [H2, 4.1] for the analogue in the pure case). Let [W]′ denote the automorphic vector bundle on Sh(Q′, X′) associated to the Q′- homogeneous vector bundle W, restricted to X̌′ ⊂ X̌. Then via the natural map (1.3.4) we have canonical isomorphisms

ϕ_{σ} : f_{Q′}^{∗}([W]_{Σ})⟶^{∼}(π′_{2,σ})^{∗}(π′_{2,∗}([W]′)^{T ′}),

and these isomorphisms are compatible with respect to inclusions σ ⊂ τ.

(iii) There is a flat automorphic vector bundle L̃ such that [W] can be realized as an automorphic subquotient of L̃, and [W]_{Σ} is a locally free subquotient of the Deligne extension L̃_{Σ}, compatibly with the given realization over Sh.

The canonical extension [W]_{Σ} exists and is unique.

Proof. The existence of [W]_{Σ} satisfying (iii) is proved in exactly the same way as the corresponding assertion for pure Shimura varieties [H2, Theorem 4.2]. The equivalence of (iii) with (ii), similarly, is identical to the proof of (4.2.2) of [H2].

Let B be as in (i). It follows from Lemma (1.3.2,ii) that, if [W]_{Σ} satisfies (iii), then there is an inclusion

(1.3.6)  Γ(B, [W]_{Σ}) ⊂ {s ∈ Γ(B ∩ Sh, [W])|s is slowly increasing}.

But then every s ∈ Γ(B ∩ Sh, [W]) in the right-hand side of (1.3.6) has a removable singularity along ∂Sh_{Σ} and therefore extends to an element of Γ(B, [W]_{Σ}). This shows the equivalence of (iii) with (i). Uniqueness is clear from (i).

As mentioned at the beginning of this section, unless an explicit statement is made to the contrary, we now assume our partial compactifications Sh_{Σ} to be compact, as well as SNC. Then ∂Sh_{Σ} is a complete divisor with normal crossings on Sh_{Σ}, defined by the invertible sheaf of ideals I_{∂Sh_{Σ}}. If [W] is an automorphic vector bundle on Sh with canonical extension [W]_{Σ}, we define the subcanonical extension of [W] to be

[W]^{sub} = [W]_{Σ}^{sub} = [W]_{Σ} ⊗ I_{∂Sh_{Σ}},

the subscript _{Σ} will be omitted whenever possible. Similarly, we will write [W]^{can} in place of [W]_{Σ} when Σ is understood.

The theory of coherent cohomology developed in [H3, H4] extends without change to the case of mixed Shimura varieties. We state the main results in the present context.

Proposition (1.3.7). Let (Q, X) be a mixed Shimura datum, and let [W] be an automorphic vector bundle on Sh = _{K_{f}} Sh = _{K_{f}} Sh(Q, X), for some open compact subgroup K_{f} of Q(A_{f}). Let Sh_{Σ} be a toroidal compactification of Sh.

(i) Let Σ_{1} be a refinement of Σ, and let f_{Σ_{1},Σ} : Sh_{Σ_{1}} → Sh_{Σ} be the corresponding morphism of compactifications. Then there are isomorphisms

f_{Σ_{1},Σ}^{∗}([W]_{Σ})⟶^{∼}[W]_{Σ_{1}}, f_{Σ_{1},Σ}^{∗}([W]_{Σ}^{sub})⟶^{∼}[W]_{Σ_{1}}^{sub},

functorial in W, and the natural morphisms

H^{q}(Sh_{Σ}, [W]_{Σ}) → H^{q}(Sh_{Σ_{1}}, [W]_{Σ_{1}}),
H^{q}(Sh_{Σ}, [W]_{Σ}^{sub}) → H^{q}(Sh_{Σ_{1}}, [W]_{Σ_{1}}^{sub})

are isomorphisms for all q.

(ii) For each q, we define

H̃^{q}(Sh, [W]^{can}) = H^{q}(Sh_{Σ}, [W]_{Σ})

for any Σ; we can also write

H̃^{q}(Sh, [W]^{can}) = \underrightarrow{lim}_{Σ} H^{q}(Sh_{Σ}, [W]_{Σ}),

the limit taken over the system of all Σ’s, all maps being isomorphisms. Similarly, we define

H̃^{q}(Sh, [W]^{sub}) = \underrightarrow{lim}_{Σ} H^{q}(Sh_{Σ}, [W]_{Σ}^{sub}),

Let

H̃^{q}([W]^{can}) = \underrightarrow{lim}_{K_{f}} H̃^{q}(_{K_{f}} Sh, [W]^{can}),

H̃^{q}([W]^{sub}) = \underrightarrow{lim}_{K_{f}} H̃^{q}(_{K_{f}} Sh, [W]^{sub}).

Then H̃^{q}([W]^{can}) and H̃^{q}([W]^{sub}) are naturally admissible Q(A_{f})-modules.

The proof of Proposition 1.3.7 follows word for word the proofs of Propositions 2.4 and 2.6 of [H4]. The action of Q(A_{f}) arising in (ii) is defined as in [H4, (2.5.2)]: to any h ∈ Q(A_{f}) and any fixed toroidal compactification_{K_{f}} Sh(Q, X)_{Σ} at level K_{f}, there is an analogous compactification _{K_{f}^{h}}Sh(Q, X)_{Σ^{h}} of _{K_{f}^{h}}Sh(Q, X), where K_{f}^{h} = h^{−1}K_{f}h. Right-multiplication by h defines an isomorphism_{K_{f}} Sh(Q, X)_{Σ}⟶^{∼}_{K_{f}^{h}}Sh(Q, X)_{Σ^{h}} that respects automorphic vector bundles and canonical extensions. By functoriality, we thus obtain an operator t_{h} in the limit on H̃^{q}([W]^{can}) and H̃^{q}([W]^{sub}).

For any automorphic vector bundle [W] on Sh we consider the Dolbeault complex A^{0,•}([W]) on Sh. For each q, A^{0,q}([W]) will denote the sheaf of C^{∞}-sections of the bundle Ω^{0,q} ⊗ [W] of (0, q)-forms with values in [W]. Letting j_{Σ} denote the embedding Sh ↪ Sh_{Σ}, we have, for each q, the direct image sheaf j_{Σ,∗}(A^{0,q}([W])) on Sh_{Σ}. We let A^{0,q}([W]))_{si} ⊂ j_{Σ,∗}(A^{0,q}([W])) denote the subsheaf of sections s such that s and \overline{∂}s are both slowly increasing, in the sense of (1.3.3), with respect to admissible metrics on the bundles Ω^{0,q} ⊗ [W] and Ω^{0,q+1} ⊗ [W]. Similarly, we define the rapidly decreasing sections, and the sheaf A^{0,q}([W]))_{rd}, as in [HZ1, 3.8.2]. We let C^{0,q}([W])_{∗} = \underrightarrow{lim}_{K_{f} ,Σ} Γ(Sh_{Σ}, A^{0,q}([W]_{∗}), where * denotes either si or rd. Theorem (1.3.8). There is a natural commutative diagram of Q(A_{f})-modules, where the horizontal maps are isomorphisms:

H̃^{•}([W]^{sub}) ⟶^{∼} H^{•}(C^{0,•}([W])_{rd})
↓ ↓
H̃^{•}([W]^{can}) ⟶^{∼} H^{•}(C^{0,•}([W])_{si})

The proof is identical to that of [H4, Corollary 3.4].

A purely algebraic construction of the canonical extension of an automorphic vector bundle was given in [HZ1], in the case of pure Shimura varieties. The analogous construction works here as well. Proposition (1.3.9). We retain the notation of Proposition (1.3.7).

(i) The standard principal bundle I(Q, X) extends to an E(Q, X)-rational Q(A_{f})- equivariant principal Q-bundle I(Q, X)_{Σ} over Sh(Q, X)_{Σ}.

(ii) The morphism p : I(Q, X) → X̌ extends to a Q-equivariant morphism p_{Σ} : I(Q, X)_{Σ} → X̌, rational over E(Q, X).

(iii) For any automorphic vector bundle [W] on Sh(Q, X), there is a canonical isomorphism of vector bundles over Sh(Q, X)_{Σ}:

[W]_{Σ}⟶^{∼}p_{Σ}^{∗}(W)/Q.

This isomorphism is rational over the field of definition of W as a Q-equivariant vector bundle over X̌. Over Sh(Q, X) this isomorphism restricts to the construction of [W] given in §1.2.

Proof. The proof is identical to that of Lemma 4.4.2 of [HZ1].

(1.3.10) Remark. Just as in Remark (1.2.15), we can formulate a conjecture for the action of Aut(ℂ) on the canonical extensions of automorphic vector bundles. It should again be a simple matter to reduce this conjecture to the case of pure Shimura varieties, where it is known [BHR, Proposition 1.4.3].

(1.4) Functorial properties of canonical extensions. We retain the notation of the previous sections, and let π_{2} : Sh(Q, X) → Sh(Q/U, U\X) and π_{1} : Sh(Q/U, U\X) → Sh(G, W \X) be the morphisms of (1.1.5). Let K_{f} ⊂ Q(A_{f}) be any neat open compact subgroup and let K_{f}^{2} = K_{f}/(U(A_{f})∩K_{f}) ⊂ (Q/U)(A_{f}), K_{f}^{1} = K_{f}/(W (A_{f})∩ K_{f}) ⊂ G(A_{f}). Then there are natural morphisms at level K_{f}, also denoted π_{2} and π_{1}:

(1.4.1)  _{K_{f}} Sh(Q, X) ⟶^{π_{2}} _{K_{f}^{2}}Sh(Q/U, U\X) ⟶^{π_{1}} _{K_{f}^{1}}Sh(G, W \X).

Now we can find families of fans Σ, Σ_{2}, Σ_{1}, admissible relative to K_{f}, K_{f}^{2}, and K_{f}^{1}, respectively, such that _{K_{f}} Sh(Q, X)_{Σ} is compact and such that the morphisms in (1.4.1) extend to the corresponding toroidal compactifications [P, 6.7(b)], which we assume to be SNC and rational over E(Q, X):

(1.4.2)  _{K_{f}} Sh(Q, X)_{Σ} ⟶^{π_{2,Σ}} _{K_{f}^{2}}Sh(Q/U, U\X)_{Σ_{2}} ⟶^{π_{1,Σ}} _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}.

In what follows, we let W be a Q-homogeneous vector bundle on X̌ and let [W] be the corresponding automorphic vector bundle on _{K_{f}} Sh(Q, X).

Proposition (1.4.3). With the above notation, the higher direct images R^{q}π_{2,Σ,∗}[W]^{can} = 0, R^{q}π_{2,Σ,∗}[W]^{sub} = 0 for q > 0, and we have canonical isomorphisms

(1.4.3.1)  π_{2,Σ,∗}[W]^{can}⟶^{∼}(π_{2,Σ,∗}[W])^{can}⟶^{∼}[W_{U}]^{can}
(1.4.3.2)  π_{2,Σ,∗}[W]^{sub}⟶^{∼}(π_{2,Σ,∗}[W])^{sub}⟶^{∼}[W_{U}]^{sub},

in the notation of (1.2.14). Moreover, there is a canonical isomorphism

(1.4.3.3)  [W]^{can}⟶^{∼}π_{2,Σ}^{∗}(π_{2,Σ,∗}[W])^{can}.

Proof. The isomorphism (1.4.3.3) is analogous to (i) of [HZ1, Proposition 3.12.2] and is proved in the same way. The vanishing of the higher direct images of [W]^{can}, together with the first isomorphism in (1.4.3.1), is a mild generalization of (ii) of [HZ1, Proposition 3.12.2]. The proof in the present situation is identical, the basic local ingredient being a calculation [HZ1, Lemma 1.6.8 (iii)] in the setting of proper morphisms of torus embeddings. The second isomorphism in (1.4.3.1) is clear from (1.2.14).

The assertions for subcanonical extensions follow as in [HZ1, (3.14.1)] from the toric version proved in [HZ1, Lemma 1.6.8 (iii)].

The following proposition is more substantial. In what follows, we let 𝔳 = Lie(V). Any x ∈ X, determines a subalgebra F_{x}^{0}(Lie(Q)) ⊂ Lie(Q), as in (1.1), and we let 𝔳_{x}^{−} = F_{x}^{0}(Lie(Q)) ∩ 𝔳, as in [HZ1, 1.8]. The (abelian) Lie algebra 𝔳_{x}^{−} depends only on the image π̌_{1} ∘ π̌_{2} of x in W \X, which we denote \overline{x} in the following proposition.

Proposition (1.4.4). Let V be a Q/U-homogeneous vector bundle on U\X̌ and let [V] be the corresponding automorphic vector bundle on _{K_{f}^{2}}Sh(Q/U, U\X). For any integer q, let V^{(q)} be the G-homogeneous vector bundle on W \X̌ ≃ (W \X)^{∨} whose fiber at the point \overline{x} is the representation of F 0 ; \overline{x}(Lie(G)) on H^{q}(𝔳_{x}^{−}, V_{x}), for any x ∈ X̌. Then for all q, there are canonical isomorphisms

(1.4.4.1)  R^{q}π_{1,Σ,∗}[V]^{can}⟶^{∼}(R^{q}π_{1,Σ,∗}[V])^{can}⟶^{∼}[V^{(q)}]^{can}
(1.4.4.2)  R^{q}π_{1,Σ,∗}[V]^{sub}⟶^{∼}(R^{q}π_{1,Σ,∗}[V])^{sub}⟶^{∼}[V^{(q)}]^{sub}

of vector bundles on _{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}. The family of such isomorphisms, as V varies, is rational over E(Q, X); in other words, the isomorphisms respect canonical models.

Proof. We use the characterization of the canonical extensions by growth conditions in (1.3.5(i)), and the calculation of their cohomology by the Dolbeault complex of forms of moderate growth (see (1.3.8); also cf. §2 below). The first isomorphism in (1.4.4.1) is a slight generalization of Proposition 3.13.4(i) of [HZ1], and is proved by the same argument. The second isomorphism, which incorporates the rationality assertion, is proved by the argument used to prove Lemma 4.7.14 of [HZ1].

Likewise, the subcanonical extension [V]^{sub} is described by conditions of rapid decrease at the boundary. The proof of Proposition 3.13.4(i) of [HZ1] works just as well for rapidly decreasing sections, yielding the first isomorphism of (1.4.4.2). The second isomorphism then follows from the second isomorphism of (1.4.4.1).

Corollary (1.4.5). Write π_{Σ} = π_{1,Σ} ∘ π_{2,Σ}. There are spectral sequences:

(1.4.5.1) \begin{matrix}E_{2}^{p,q} = H^{p}(_{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}, & (R^{q}π_{Σ,∗}[W])^{can}) \\ ⇒ H^{p+q}(_{K_{f}^{2}}Sh(Q/U, U\X)_{Σ_{2}}, [W_{U}]^{can}) \\ ⟶^{∼} H^{p+q}(_{K_{f}} Sh(Q, X)_{Σ}, [W]^{can}); \end{matrix}

(1.4.5.2) \begin{matrix}E_{2}^{p,q} = H^{p}(_{K_{f}^{1}}Sh(G, W \X)_{Σ_{1}}, & (R^{q}π_{Σ,∗}[W])^{sub}) \\ ⇒ H^{p+q}(_{K_{f}^{2}}Sh(Q/U, U\X)_{Σ_{2}}, [W_{U}]^{sub}) \\ ⟶^{∼} H^{p+q}(_{K_{f}} Sh(Q, X)_{Σ}, [W]^{sub}). \end{matrix}

Suppose [W] satisfies a Kostant condition (see (1.2.16)). Then these spectral sequences degenerate at E_{2} and the filtration induced by the E_{2} term splits canonically.

Proof. Given Propositions 1.4.3 and 1.4.4, the first part is just the Leray spectral sequence for π_{Σ}. When [W] satisfies a Kostant condition, the reason for degeneration at E_{2} and the splitting are given in (1.2.16.3). To elaborate further, the prototype of the use of the Kostant condition is given in [HZ1,Corollary 3.7.8], the version of the present corollary for unextended automorphic vector bundles. The degeneration of the spectral sequences for canonical and subcanonical extensions can be found in [HZ1, 3.13–3.14]. These results were obtained by following the proof of [HZ1, Corollary 3.7.8], paying close attention to growth and decay conditions. Proposition 1.4.4 ensures that exactly the same arguments work here. As noted in Remark (1.2.16.3), The importance of (1.2.16.2) in proving the splitting was noted in Remark (1.2.16.3).

(1.4.6) Periods and rationality. Mimicking sections 4.7 and 4.8 of [HZ1], we show that the isomorphisms of Propositions (1.4.3) and (1.4.4) are compatible with the canonical rational structures described in (1.2.A). Let π : X → W \X be the natural map. For j ≥ 0 we define the homogeneous vector bundle W_{U′}^{(j)} on W \X as in the previous section; its fiber at the point π(x) ∈ W \X, for x ∈ X, is given by

(1.4.6.1)  W_{x}^{(j)} =^{def} H^{j}(𝔳_{x}^{−}, W_{x}),

Let σ_{x,W}^{j} denote the representation of Stab_{W \Q}(π(x)) on W_{x}^{(j)}. The following lemma is the analogue of Lemma 4.7.14 of [HZ1], and is proved in the same way:

Lemma (1.4.6.2). Let (T, h) ↪ (Q, X) be a CM pair, as above. Let L ⊃ E(T, h; W) and t_{f} ∈ T (A_{f}). Then we have

p(h, σ_{x,W}^{j}) · Per_{(h,t_{f})}W_{x}^{(j)}(L) = R^{j}π_{∗}[W](L)_{[h,t_{f}]} ⊂ R^{j}π_{∗}[W]|_{[h,t_{f}]}(ℂ).

(1.5) A boundary calculation. In the present section, we generalize Proposition 1.4.5 to a situation arising from the study of the boundary cohomology of pure Shimura varieties. The length of the calculation is primarily due to the complexity of the requisite notation.

Thus, let (G, X) be a pure Shimura datum, which we may as well assume to be standard (i.e., the map h of (1.1.2) is a bijection). For simplicity, we assume G^{ad} to be ℚ-simple. Then the standard maximal ℚ-parabolics of G are totally ordered as in (0.7).

Let P, P′ be two standard maximal ℚ-parabolics, with P′ < P. Let P (P′) = P′∩G_{h,P}; then P (P′) is a maximal proper parabolic subgroup of G_{h,P}, hence corresponds to a boundary stratum in the toroidal compactification of Sh(G_{h,P} , X(P)), where X(P) is as in (0.5). Let Q(P) ⊂ P denote the canonical subgroup, in the terminology of [P,§4]: if L(P) = G_{h,P} ·G_{ℓ,P} is viewed as a (standard) Levi subgroup of P, then Q(P) = R_{u}P · G_{h,P}, with associated mixed Shimura datum (Q(P), X (P)). We define the mixed Shimura data (Q(P′), X (P′)) and (Q(P (P′)), X (P (P′))) analogously. We write

Q′ = Q(P′), Q = Q(P), Q′′ = Q(P (P′)); X′ = X (P′), X = X (P), X′′ = X (P (P′)) = W \X′,

and let

U = U_{Q} = W_{−2}Q, U′ = U_{Q′} = W_{−2}P′;
W = W_{Q} = R_{u}Q(= R_{u}P), W′ = W_{Q′} = (W ∩ W′) · U′.

Note that G_{h,P (P′)} = G_{h,P′} and U ⊂ U′.

Let P_{1} = P (P′) · W denote the admissible parabolic of Q corresponding to the parabolic subgroup P (P′) ⊂ G_{h,P}. Then Q′ ⊂ P_{1}; indeed, Q′ is precisely the canonical subgroup of P_{1} (cf. [P], Lemma 4.19; note that our P ’s are Pink’s Q’s and vice versa). In particular, W ∩ U′ ⊂ W′, U ⊂ U′ ∩ W, Q′′ = Q′/(W ∩ W′). Let

W′′ = W_{Q′′} = W′/(W ∩ W′), U′′ = U_{Q′′} = U′/(U′ ∩ W).

Let V = W/U, V′ = W′/U′, and define 𝔳_{x}^{−} = F_{x}^{0}(Lie Q)∩𝔳, 𝔳_{x}^{′,−} = F_{x}^{0}(Lie Q)∩𝔳′ as in the paragraph preceding Proposition 1.4.4. Let V_{x}^{−} ⊂ V and V_{x}^{′,−} ⊂ V′ be the corresponding subgroups. Let Q_{P,x} (resp. Q_{P′,x}) be the maximal parabolic subgroup of K_{x} (denoted Q_{F,x} in [HZ1, Lemma 1.8.6]) with unipotent radical V_{x}^{−} (resp. V_{x}^{′,−}). Let K_{h} = K_{x} ∩ G_{h,P}, K′_{h} = K_{x} ∩ G_{h,P′} (cf. [HZ1, 1.8]).

The adjoint representation of G_{h,P} on 𝔳 endows the latter with a variation of Hodge structure of weight −1 over X(P), corresponding to the homology in degree one of the abelian scheme Sh(Q/U, U\X) over Sh(G_{h,P} , X(P)). (This is the abelian scheme denoted A_{P} in [HZ2]; cf. (2.5), below). The maximal parabolic subgroup P (P′) ⊂ G_{h,P} corresponds to a degeneration to a mixed Hodge structure, with weight filtration denoted W_{•}^{P (P′)}𝔳, (denoted W_{•}^{F} 𝔳 in [HZ1, 1.2.2]). We write Gr_{i}^{P (P′)}𝔳 = W_{i}^{P (P′)}V/W_{i−1}^{P (P′)}V. Let 𝔳_{x}^{′′,−} = F_{x}^{0}(Lie Q) ∩ Gr_{−1}^{P (P′)}𝔳.

Let [W] denote a fully decomposed automorphic vector bundle on the original pure Shimura variety Sh(G, X). Thus for any point x ∈ X, with stabilizer K_{x} ⊂ G, [W] is the locally homogenous holomorphic vector bundle associated to some algebraic representation λ_{x} : K_{x}→GL(W_{x}). In the calculation in [HZ1] of the part of the boundary cohomology of [W] corresponding to P, we associated to [W] an automorphic vector bundle [W(P)] (which would be [W(F)] in the notation of [HZ1,4.7]) on Sh(Q, X). Concretely, [W(P)] is the locally homogenous holomorphic vector bundle associated to λ_{x} |_{K_{h,P} ·V_{x}^{−}}. Note that Q_{P,x} maps isomorphically to its image in Q/U, and the representation λ_{x} defines the bundle [W(P)_{U}] (cf. (1.2.14)) on the abelian scheme Sh(Q/U, U\X).

Let Sh(Q, X) ↪ Sh(Q, X)_{Σ} be a (SNC) toroidal compactification. We will be studying the coherent cohomology of the pullback of the canonical extension [W(P)]^{can} to the part of the boundary of Sh(Q, X)_{Σ} corresponding to the admissible parabolic P_{1}. Let D_{ℓ,Q′} denote the quotient P_{1}/Q′. This is the group that plays the role of G_{ℓ,P′} in Pink’s theory of the toroidal compactification of Sh(Q, X) (and if Q were reductive, it would reduce to G_{ℓ,P′}); we call it the D_{ℓ}-group in what follows. Its unipotent radical R_{u}D_{ℓ,Q′} equals W/(W ∩ W′), with Lie algebra Gr_{0}^{P (P′)}𝔳.

Let (P ∩ P′)_{ℓ,P′} = P ∩ G_{ℓ,P′}, a maximal parabolic subgroup of G_{ℓ,P′}. (this is consistent with the notation to be used in §3). Then it is easy to check that

(1.5.1)  R_{u}D_{ℓ,Q′} = R_{u}(P ∩ P′)_{ℓ,P′}.

Moreover, the standard Levi subgroup of (P ∩ P′)_{ℓ,P′} is naturally identified with G_{ℓ,P} · G_{ℓ,P (P′)}. Let P_{1,ℓ} = G_{ℓ,P (P′)} · R_{u}(P ∩ P′)_{ℓ,P′}. Then P_{1} is the subgroup P_{1,ℓ} · G_{h,P′} · W′ ⊂ P, and there are natural isomorphisms

(1.5.2) D_{ℓ,Q′} ⟶^{∼}P_{1,ℓ}/(G_{h,P′} ∩P_{1,ℓ})⟶^{∼}G_{ℓ,P (P′)}/(G_{ℓ,P (P′)} ∩G_{h,P′})·R_{u}(P ∩P′)_{ℓ,P′}.

Note that

(1.5.3) Lie(Q_{P,x} ∩ Q′) = Lie(K′_{h}) ⊕ 𝔳_{x}^{′′,−}; Lie(Q_{P′,x} ∩ Q′) = Lie(K′_{h}) ⊕ 𝔳_{x}^{′,−}.

In particular, Q_{P′,x} ∩ Q′ contains the unipotent radical of Q_{P′,x}. On the other hand, it follows easily from (1.5.1) that Cayley transform defines an isomorphism

(1.5.4)  𝔳_{x}^{−} = Lie R_{u}Q_{P,x} ⟶^{∼} Lie R_{u}D_{ℓ,Q′} ⊕ 𝔳_{x}^{′′,−}.

The choice of a base point x is made only for convenience in characterizing automorphic vector bundles, and plays no essential role in the theory. In what follows, the subscript x will be used uniformly to designate a point of any of the homogeneous spaces X, X′, X′′, etc., the sole condition being that the various x’s be chosen compatibly with respect to morphisms of mixed Shimura data.

In addition to (Q/U, U\X), and (G_{h,P} , X(P)), we will be working with the intermediate Shimura data (Q′/U, U\X′) and (Q′/W, W \X′) = (Q′′, X′′) There are three more relevant Shimura data: (Q′/U′, U′\X′), (Q′′/U′′, U′′\X′′), and (Q′/W′, W′\X′) = (G_{h,P′}, X(P′)). We write _{•} where the level subgroups would be indicated. We find three sequences of morphisms

(1.5.5)  _{•}Sh(Q′, X′) ⟶^{π′_{2}} _{•}Sh(Q′/U′, U′\X′) ⟶^{π′_{1}} _{•}Sh(G_{h,P′}, X(P′))
(1.5.6)  _{•}Sh(Q′′, X′′) ⟶^{π′′_{2}} _{•}Sh(Q′′/U′′, U′′\X′′) ⟶^{π′′_{1}} _{•}Sh(G_{h,P′}, X(P′))
(1.5.7)  _{•}Sh(Q′, X′) ⟶^{π(P′)_{2}} _{•}Sh(Q′/U, U\X′) ⟶^{π(P′)_{1}} _{•}Sh(Q′′, X′′)

The first two are instances of boundary diagram (1.1.5), whereas (1.5.7) is the boundary diagram for the mixed Shimura variety Sh(Q, X) corresponding to the admissible parabolic subgroup P_{1} ⊂ Q.

We choose compatible families of fans Σ′, Σ′_{2}, Σ′_{1} (resp. Σ′′, Σ′′_{2}, Σ′′_{1} = Σ′_{1}, resp. Σ′, Σ_{2}, Σ_{1}) so that the morphisms in diagram (1.5.5) (resp. (1.5.6), resp. (1.5.7)) extend to morphisms of toroidal compactifications. By definition we have Σ = ∪{Σ_{P_{α}}} where P_{α} runs through the standard admissible parabolic subgroups of Q, and likewise for the other data. However, to avoid irrelevant notation we designate the toroidal compactification of any of the Shimura varieties ? simply by ?_{Σ}. Again, note that the data for (1.5.7) represent part of the data for the toroidal compactification of Sh(Q, X). In the end, we obtain a large commutative diagram

(1.5.8)

•Sh(Q′, X′)_{Σ}
π(P′)_{2,Σ}↓
•Sh(Q′/U, U\X′)_{Σ}
ψ_{1}↓
•Sh(Q′/U′ ∩ W, (U′ ∩ W)\X′)_{Σ} ⟶^{ψ_{2}} _{•}Sh(Q′/U′, U′\X′)_{Σ}
ψ_{3}↓  ^{ψ_{4}}↓
•Sh(Q′′, X′′)_{Σ}  ⟶^{π′′_{2,Σ}} _{•}Sh(Q′′/U′′, U′′\X′′)_{Σ}
π′′_{1,Σ}↓
•Sh(G_{h,P′}, X(P′))_{Σ}

Here the arrows ψ_{i}, i = 1, … 4, have been introduced to make the diagrams commute. We then have

(1.5.9)  ψ_{2} ∘ ψ_{1} ∘ π(P′)_{2,Σ} = π′_{2,Σ}; π′′_{1,Σ} ∘ ψ_{4} = π′_{1,Σ}; ψ_{3} ∘ ψ_{1} = π(P′)_{1,Σ}

One checks that ψ_{1} and ψ_{2} are compactifications of toric fibrations, whereas ψ_{3} and ψ_{4} are generically abelian schemes. The left-hand column in (1.5.8), a refinement of (1.5.7) will be analyzed below in connection with the Q′-part of the boundary in the compactification of the diagram (1.1.5):

(1.5.10)  _{•}Sh(Q, X)_{Σ} ⟶^{π_{2}} _{•}Sh(Q/U, U\X)_{Σ} ⟶^{π_{1}} _{•}Sh(G_{h,P} , W \X)_{Σ}.

At no point in our calculation of boundary coherent cohomology have we seriously used the fact that W is the full unipotent radical of Q. The only real restriction is the Kostant condition, used to prove splitting in Proposition 1.4.5. We now apply the analysis of [HZ1,§3] to study the boundary cohomology of [W(P)]. We begin by choosing a connected component D ⊂ X and replacing the disconnected (finite-level) Shimura varieties _{•}Sh(Q, X) by a connected component of the form M_{Γ} = Γ\D. As above, we use the subscript Σ uniformly for all (compatible) toroidal compactifications. Recall that M_{Γ} has a Baily-Borel compactification M_{Γ}^{∗} [P, 6.2], defined set-theoretically as the disjoint union of boundary strata M_{Γ}(P_{α}) for Γ-conjugacy classes of rational parabolic subgroups P_{α} of Q. There is a surjective map ϕ_{Σ} : M_{Γ,Σ}→M_{Γ}^{∗}. We let Z_{Q′} ⊂ M_{Γ,Σ} denote the closure of the subset ϕ_{Σ}^{−1}M_{Γ}(P_{1}) (see [HZ1,(1.5.1)] for the analogue in the pure setting). Under our hypotheses, it is a divisor with normal crossings, each of whose components is smooth. We denote by i_{Q′} the closed immersion Z_{Q′} ↪ M_{Γ,Σ}.

As in the pure case (cf. [HZ1,1.3]), the fan Σ_{P_{1}} is a polyhedral decomposition of a certain subset of the closure of a cone C(Q′) ⊂ U′(ℝ)(−1) (the Tate twist is as in [P]), homogeneous with respect to P_{1}(ℝ)^{0} and invariant under translation by (U′ ∩ W)(ℝ)(−1). Pink proves that the quotient C(Q′)/(U′ ∩ W)(ℝ)(−1) is a non-degenerate homogeneous cone for G_{ℓ,P (P′)}(ℝ)^{0} in U_{P (P′)}(ℝ)(−1) (for all this, see [P], Proposition 4.15). The fan Σ_{P_{1}} is equivariant with respect to an appropriate congruence subgroup Δ_{ℓ,Q′} ⊂ D_{ℓ,Q′}(ℚ) (denoted Δ_{1} by Pink in its adelic incarnation). Our running hypothesis that Γ be neat (cf. (1.3)) implies that Δ_{ℓ,Q′} acts freely on Σ_{P_{1}}. We may assume without loss of generality that Δ_{ℓ,Q′} admits a factorization as semi-direct product

(1.5.11)  Δ_{ℓ,Q′} ⟶^{∼}Γ_{ℓ,P (P′)} × Γ_{u},

where Γ_{ℓ,P (P′)} is the image in G_{ℓ,P (P′)}/(G_{ℓ,P (P′)} ∩ G_{h,P′})(ℚ) of a congruence subgroup Γ^{1} of G_{ℓ,P (P′)}(ℚ) and Γ_{u} is a lattice in the vector group R_{u}D_{ℓ,Q′}. (The hypothesis that Γ be neat implies that the map Γ^{1} → Γ_{ℓ,P (P′)} is an isomorphism.)

We define Σ_{P_{1}}^{c} and Σ̂_{P_{1}}^{c} as in [HZ1, (1.3.2)] and [HZ1,(2.2.2)], respectively. The irreducible components of Z_{Q′} are indexed by the one-dimensional faces in Σ_{P_{1}} modulo the action of Δ_{ℓ,Q′}. Then it is easy to verify that the analogue of Lemma 3.7.2 of [HZ1] remains true in the present setting:

Lemma (1.5.12). The nerve 𝔑(Z_{Q′}) of the closed covering of Z_{Q′} by its irreducible components is isomorphic to Δ_{ℓ,Q′}\Σ̂_{P_{1}}^{c}.

As before, we write _{•} where the level subgroups belong, to avoid proliferation of notation. We now consider the restriction of (1.5.10) to the connected component M_{•,Σ}

(1.5.13)  M_{•,Σ} ⟶^{π_{2}} A_{Q/U,•,Σ} ⟶^{π_{1}} M_{P,•,Σ};

here, A_{Q/U,•,Σ} (resp. M_{P,•,Σ}) is a connected component of a toroidal compactification of _{•}Sh(Q/U, U\X) (resp. _{•}Sh(G_{h,P} , X′′)). It follows from Pink’s theory that A_{Q′/U,•,Σ} is a toroidal compactification of a semi-abelian scheme over M_{Q′′,•,Σ}. There is a commutative diagram:

(1.5.14)  \begin{matrix}Z_{Q′} & ⟶^{i_{Q′}} & M_{•,Σ} \\ ^{π(P′)_{2}}↓ & & ^{π_{2}}↓ \\ Z_{Q′/U} & ⟶^{i_{Q′/U}} & A_{Q/U,•,Σ} \\ ^{π(P′)_{1}}↓ & & ^{π_{1}}↓ \\ Z_{Q′′} & ⟶^{i_{Q′′}} & M_{P,•,Σ} \end{matrix}

As in the discussion of Σ_{P_{1}} above, the second and third lines of (1.5.14) correspond to fans Σ_{P_{1}/U} ⊂ C(Q′/U) ⊂ U′/U(ℝ)(−1) and Σ_{P_{1}/W} ⊂ C(Q′′) ⊂ (U′/U′ ∩ W)(ℝ)(−1), respectively, with the obvious notation. The corresponding D_{ℓ}-groups are denoted D_{ℓ,Q′/U} and D_{ℓ,Q′′}. It is easy to see that there are maps

Σ_{P_{1}} → Σ_{P_{1}/U} → Σ_{P_{1}/W} ,

where the first map is a bijection, and that

(1.5.15) D_{ℓ,Q′} ⟶^{∼}D_{ℓ,Q′/U}; D_{ℓ,Q′′} ⟶^{∼}D_{ℓ,Q′/U}/R_{u}D_{ℓ,Q′/U} ⟶^{∼}G_{ℓ,P (P′)},

C(Q′/U)⟶^{∼}C(Q′)/U(ℝ)(−1); C(Q′′)⟶^{∼}C(Q′)/(U′ ∩ W)(ℝ)(−1).

The two lines of (1.5.15) are compatible. As an immediate consequence, we have the following elaboration on Lemma (1.5.12):

Lemma (1.5.16). Define 𝔑(Z_{Q′/U}) and 𝔑(Z_{Q′′}) to be the nerves of the closed coverings of Z_{Q′/U} and Z_{Q′′}, respectively, by their irreducible components. Then there is a diagram

𝔑(Z_{Q′}) ⟶^{𝔑(π(P′)_{2})} 𝔑(Z_{Q′/U}) ⟶^{𝔑(π(P′)_{1})} 𝔑(Z_{Q′′})

where the first arrow is an isomorphism and the second is homotopy equivalent to a fibration with fiber Γ_{u}\[(U′ ∩ W/U)(ℝ)(−1)].

To each σ ∈ Σ_{P_{1}} corresponds an irreducible closed stratum Z_{σ} ⊂ Z_{Q′}, a toroidal compactification of a connected component of the mixed Shimura variety_{•}Sh(Q′/U(σ), X′), where U(σ) ⊂ U′ is the rational subgroup whose group of real points is generated by σ. The left-hand column of (1.5.14), restricted to Z_{σ}, is a diagram of connected components for the analogue of (1.5.7) where Q′ is replaced by Q′/U(σ). Fix any such Z_{σ}. Then the left hand column of (1.5.8) gives rise to a diagram of connected components:

(1.5.17)  Z_{σ} ⟶^{π(P′)_{2}} A_{Q′/(U+U(σ)),•,Σ} ⟶^{ψ_{1}} A_{Q′/(U′∩W +U(σ)),•,Σ} ⟶^{ψ_{3}} Z_{σ′′}.

Here σ′′ denotes the image of σ in C(Q′′), and Z_{σ′′} = M_{Q′′/U(σ′′),•,Σ} is the corresponding closed stratum in Z_{Q′′}. As σ′′ varies, the diagrams (1.5.17) fit together to a refinement of the left-hand column of (1.5.14):

(1.5.18)  Z_{Q′} ⟶^{π(P′)_{2}} Z_{Q′/U} ⟶^{ψ_{1}} A_{Q′/U′∩W} ⟶^{ψ_{3}} Z_{Q′′}.

This can be continued with a map π′′ : Z_{Q′′}→M_{P′,•,Σ}, where M_{P′,•,Σ} is a connected component of Sh(G_{h,P′}, X(P′))_{Σ}, that is the toroidal compactification of (1.5.6).

Fix σ′′ ⊂ C(Q′′), and let Z_{Q′}(σ′′) ⊂ Z_{Q′} denote the union of the strata Z_{σ} that map to Z_{σ′′} under π(P′) := π(P′)_{1} ∘ π(P′)_{2}. Lemma 1.5.16 and the discussion below diagram (1.5.8) imply that (1.5.19) The nerve 𝔑(Z_{Q′}(σ′′)) of the closed covering of Z_{Q′}(σ′′) by its irreducible

components is homotopy equivalent to the compact torus Γ_{u}\[(U′ ∩W/U)(ℝ)(−1)] (more precisely, it is isomorphic to a triangulation of the latter).

(1.5.20) Over π^{′′,−1}(M_{P′,•}) (i.e., over the open stratum), ψ_{3} is an abelian scheme whose typical anti-holomorphic tangent space is 𝔳_{x}^{′′,−}; in particular, it is a bijection on sets of irreducible components.

(1.5.21) π(P′)_{2} is a bijection on sets of irreducible components.

By Proposition (1.4.3), we have canonical isomorphisms

(1.5.22)  \begin{matrix}π(P′)_{2,∗}([W(P)]^{can}) ⟶^{∼} i_{Q′/U}^{∗}[W(P)_{U}]^{can}; \\ R^{i}π(P′)_{2,∗}([W(P)]^{can}) = 0, i > 0. \end{matrix}

Similarly, there is an automorphic vector bundle that we might denote [W(P′)_{U′∩W}] on Sh(Q′/(U′ ∩ W), (U′ ∩ W)\X′′) such that, for any diagram of closed strata as in (1.5.17), the pullback to A_{Q′/(U+U(σ)),•,Σ} of i_{Q′/U}^{∗}[W(P)_{U}]^{can} is isomorphic to ψ_{1}^{∗}[W(P′)_{U′∩W}].

It then follows from (1.5.19) and (1.5.20), as in [HZ1, (3.12.6)], that

(1.5.23)  R^{q}π(P′)_{∗}(i_{Q′}^{∗}[W(P)]^{can}) ≃ H^{q}(R_{Γ_{u}}ψ_{3,∗}[W(P)_{U′∩W}]^{can});

here recall (cf. (1.5.9)) that π(P′) is the composite of the three arrows in (1.5.18). The notation R_{Γ_{u}}ψ_{3,∗} for equivariant higher direct images is as in [HZ1], e.g. [HZ1,3.12.4].

Proposition (1.5.24). The Leray spectral sequence

E_{2}^{p,q} = H^{p}(Z_{Q′′}, R^{q}π(P′)_{∗}(i_{Q′}^{∗}[W(P)]^{can})) ⇒ H^{p+q}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can})

degenerates at E_{2} and the gradation induced by the E_{2} term splits canonically.

Proof. As in the proof of Corollary 1.4.5, the point is to apply an appropriate Kostant condition, in this case to the mixed Shimura datum (Q′/U, U\X′). The group K_{x} ⊂ G stabilizing x ∈ X is the K_{x} of (1.2.16.1); for Π_{x} we take Q_{P,x}. Instead of the full subgroup Q_{x}^{0}∩(W′/U) for the mixed Shimura datum (Q′/U, U\X′), we take the subgroup Q_{x}^{0} ∩ (W/U) = Q_{x}^{0} ∩ V, whose Lie algebra is just 𝔳_{x}^{−}. The map

α_{x} : 𝔳_{x}^{−}⟶^{∼}Lie(R_{u}Q_{P,x})

is defined, as in [HZ1,1.8.6], in terms of the Cayley transform for the boundary subgroup P of G. Thus we are in the situation of (1.2.16.4). and we use the canonical splitting (1.2.16.5). Note that all choices are canonical with respect to our choice of the point x, the standard parabolics P, P′, and the standard Levi components. In this sense, the splitting we will obtain on the E_{2} term is also canonical.

We next set up the Leray spectral sequence, so that we may apply the above. The automorphic vector bundle [W(P)_{U′∩W}] is the locally homogeneous holomorphic vector bundle associated to the restriction to

Q_{P,x} ∩ Q′ ≃ (Q_{P,x} ∩ Q′)/(Q_{P,x} ∩ Q′ ∩ U′ ∩ W)

of the representation λ_{x} : K_{x}→GL(W_{x}) introduced near the beginning of (1.5). In particular, its fiber at x is W_{x}. Just as in Proposition 1.4.4,

(1.5.24.1)  R^{q}ψ_{3,∗}[W(P)_{U′∩W}]^{can} ⟶^{∼}[W(P)^{(q),′′}]^{can}

where [W(P)^{(q),′′}] is the automorphic vector bundle whose fiber at x is

(1.5.24.2)  H^{q}(Lie R_{u}(Q_{P,x} ∩ Q′), W_{x}) = H^{q}(𝔳_{x}^{′′,−}, W_{x});

the equality follows from (1.5.3).

Next, Γ_{u} acts trivially on A_{Q′/U′∩W}, hence the Γ_{u}-equivariant cohomology on the right-hand side of (1.5.23) is computed fiber by fiber. Thus H^{q}(R_{Γ_{u}}ψ_{3,∗}[W(P)_{U′∩W}]^{can}) is the automorphic vector bundle whose fiber at x is

(1.5.24.3)  ℍ^{q}(R(Γ_{u}) ∘ R(𝔳_{x}^{′′,−})(W_{x})).

Here R(Γ_{u}) and R(𝔳_{x}^{′′,−}) are abbreviations for the right-derived functors of H^{0}(Γ_{u}, •) and H^{0}(𝔳_{x}^{′′,−}, •), respectively. We use the analogous notation for Lie algebra cohomology below. Now by Nomizu’s theorem (cf. [HZ2, (5.4.7)]), there is a canonical isomorphism of functors

(1.5.24.4)  R(Γ_{u}, •) ⟶^{∼} R(Lie R_{u}D_{ℓ,Q′}, •).

It follows from (1.5.4) that the group in (1.5.24.3) is canonically isomorphic to H^{q}(𝔳_{x}^{−}, W_{x}). Let [W(Q′′)^{(q)}]^{can} be the canonically extended automorphic vector bundle on Z_{Q′′} associated to the representation of Q_{P (P′),x} on H^{q}(𝔳_{x}^{−}, W_{x}). We have thus constructed a canonical isomorphism

(1.5.24.5)  H^{q}(R_{Γ_{u}}ψ_{3,∗}[W(P)_{U′∩W}]^{can}) ⟶^{∼} [W(Q′′)^{(q)}]^{can};

(this is obvious without the canonical extensions, and that it holds with them follows from using growth conditions, as in the proof of Proposition (1.4.4).)

By (1.5.23), the Leray spectral sequence can thus be rewritten

(1.5.24.6)  E_{2}^{p,q} = H^{p}((Z_{Q′′}, [W(Q′′)^{(q)}]^{can})) ⇒ H^{p+q}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can}).

Since 𝔳_{x}^{−} is the unipotent radical of a parabolic subalgebra of K_{x}, the Kostant condition is satisfied, and the degeneration and splitting follow (as described at the beginning of the proof).

One verifies from the explicit descriptions in terms of λ_{x} that, for each q, [W(Q′′)^{(q)}]^{can} is the pullback via i_{Q′′} of the canonically extended fully decomposed automorphic vector bundle [W(P)^{(q)}]^{can} on the (connected) pure Shimura variety M_{P,•,Σ}; notation is as in Proposition (1.4.4). Applying Kostant’s theorem, Proposition (1.5.24) yields an isomorphism:

(1.5.25) H^{n}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can}) ⟶^{∼} ⨁_{b∈W^{P,x}} H^{n−ℓ(b)}(Z_{Q′′}, i_{Q′′}^{∗}[V_{λ(h,b)}]^{can}⊗V_{λ(ℓ,b)})

as in [HZ1,(3.7.4)]. Thus we let

[W(Q′′)_{λ(h,b)}]^{can} = i_{Q′′}^{∗}[V_{λ(h,b)}]^{can}.

Then we can apply the analysis of §3 of [HZ1] to the cohomology of [W(Q′′)_{λ(h,b)}] to obtain

(1.5.26) H^{s}(Z_{Q′′}, [W(Q′′)_{λ(h,b)}]^{can})

⟶^{∼} ⨁_{a∈W^{P (P′),x}} ⨁_{r} H^{s−r−ℓ(a)}(M_{P (P′),•,Σ}, [W(Q′′)_{λ(h,b);λ(h,a)}]^{can})⊗H^{r}(Γ_{ℓ,P (P′)}, V_{λ(h,b);λ(ℓ,a)}).

Here the first term λ(h, b) in the subscript corresponds to the decomposition (1.5.25), whereas the second term corresponds to the decomposition relative to the parabolic Q_{P (P′),x} ⊂ K_{h,P}. Note that M_{P (P′),•,Σ} coincides with M_{P′,•,Σ}, (compare the final terms in (1.5.5) and (1.5.6)).

Let R = P ∩ P′; the standard Levi factor of R is G_{ℓ,R} × G_{h,P′}, where G_{ℓ,R} is the standard Levi factor of R ∩ G_{ℓ,P′}. Let Q^{R} = Q_{P,x} ∩ Q_{P′,x}. This is a parabolic subgroup of K_{x} with Levi factor isomorphic via Cayley transform to G_{ℓ,R} × K_{h,P′}. Let W^{R}(= W^{R,x}) be the corresponding set of Kostant representatives in the Weyl group of K_{x}. Then (see Lemma 3.1.6 below), we can combine the decompositions (1.5.25) and (1.5.26) to write:

(1.5.27) H^{n}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can}) ⟶^{∼}

⨁_{w∈W^{R}} ⨁_{r} H^{n−ℓ(w)−r}(M_{P′,•,Σ}, [V_{λ(h,w)}]^{can}) ⊗ H^{r}(Γ_{ℓ,P (P′)}, V_{λ(ℓ,w)}).

More general results are obtained in section 3.3. The decomposition (1.5.27) can be compared with (3.3.1), below. Note, however, that the second factor on the right here is the cohomology of Γ_{ℓ,P (P′)} and not that of a congruence subgroup of G_{ℓ,R}(ℚ). Indeed, as in (3.3.4), below, we have G_{ℓ,R} = G_{ℓ,P} · G_{ℓ,P (P′)}

This is the “long way” to calculate the left-hand side of (1.5.27). The short way is to apply Corollary (1.4.5) to the different strata Z_{σ} of Z_{Q′}, and to use the analogue of diagram (1.5.5) for the mixed Shimura varieties _{•}Sh(Q′/U(σ), X′) (cf. (1.5.17)) as σ varies:

(1.5.28) _{•}Sh(Q′/U(σ), X′) ⟶^{π′_{2}} _{•}Sh(Q′/U′, U′\X′) ⟶^{π′_{1}} _{•}Sh(G_{h,P′}, X(P′)).

Note that the last two terms of (1.5.28) are independent of σ. Applying Lemma 1.5.12 directly to the configuration of the Z_{σ}, we find the following equivariant version of (1.4.5.1):

Lemma (1.5.29). There is a spectral sequence

E_{2}^{p,q} = H_{Δ_{ℓ,Q′}}^{p} (M_{P′,•,Σ}, R^{q}π′_{∗}i_{Q′}^{∗}[W(P)]^{can}) ⇒ H^{p+q}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can})

This spectral sequence degenerates at E_{2} and the filtration induced on the abutment splits canonically.

We are using the fact that i_{Q′}^{∗}[W(P)]^{can} is, on each Z_{σ}, isomorphic to the restriction to Z_{σ} of [W(P′)]^{can}. This is because _{•}Sh(Q, X)_{Σ}, can be realized as a part of a toroidal compactification of _{•}Sh(G, X), and then Z_{Q′} is part of the P′-stratum of the latter. Then the lemma follows from the characterization [HZ1, (3.2.2)] of the canonical extension on Sh(G, X)_{Σ}. The last assertions are a consequence of the Kostant condition, applied essentially as in (1.5.24) (cf. [HZ1, (3.13.6)]). In this case Π_{x} = Q_{P′,x} ⊂ K_{x}, and Lie(Q_{x}^{0}) = 𝔳_{x}^{′,−}, with the isomorphism α_{x} being given by Cayley transform as in [HZ1,(1.8.6)].

We can write the splitting in (1.5.29) in terms of Kostant representatives in W^{P′}, as above. Since Δ_{ℓ,Q′} ⊂ G_{ℓ,P′}, we obtain an isomorphism (1.5.30)

H^{n}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can}) ⟶^{∼} ⨁_{t∈W^{P′,x}} H^{n−r−ℓ(t)}(M_{P′,•,Σ}, [V_{λ(h,t)}]^{can})⊗H^{r}(Δ_{ℓ,Q′}, V_{λ(ℓ,t)}).

The last factor on the right can be computed by a Hochschild-Serre spectral sequence via (1.5.11). Recalling that Γ_{u} is a lattice in the unipotent radical of R_{ℓ,P′} := R ∩ G_{ℓ,P′}, (see (1.5.1) and applying Nomizu’s theorem and Kostant’s theorem again, we have

(1.5.31)  H^{r}(Δ_{ℓ,Q′}, V_{λ(ℓ,t)}) ⟶^{∼} ⨁_{s∈W^{R_{ℓ,P′}}} H^{r−ℓ(s)}(Γ_{ℓ,Q′}, V_{λ(ℓ,t);λ(ℓ,s)}).

Applying Lemma 3.1.6 (below) again, we can combine the last two formulas. As a special case of the formulas in Section 3.3 below, we find that, if w ∈ W^{R} equals st, with s ∈ W^{R_{ℓ,P′}} and t ∈ W^{P′,x} then

(1.5.32)  \begin{aligned}[V_{λ(h,t)}] & ≃ [V_{λ(h,w)}]; \\ V_{λ(ℓ,t);λ(ℓ,s)} & ≃ V_{λ(ℓ,w)}, \end{aligned}

where the right-hand sides are as in (1.5.27). Thus (1.5.29) gives another isomorphism

(1.5.33) H^{n}(Z_{Q′}, i_{Q′}^{∗}[W(P)]^{can}) ⟶^{∼}

⨁_{w∈W^{R}} ⨁_{r} H^{n−ℓ(w)−r}(M_{P′,•,Σ}, [V_{λ(h,w)}]^{can}) ⊗ H^{r}(Γ_{ℓ,P (P′)}, V_{λ(ℓ,w)}).

Proposition (1.5.34). The isomorphisms (1.5.27) and (1.5.33) are the same.

Proof. This comes down to the commutativity of the diagram (1.5.8). The short calculation goes by way of the maps ψ_{4} and ψ_{2}, whereas the long one involves π′′_{2,Σ} and ψ_{3} (use (1.5.9) in both cases).

(1.5.35) Remarks. (i) Proposition 1.5.34 is the point of this section. It is applied at the end of section 3.4. The key step is the proof of Proposition 1.5.24, where the Kostant condition is obtained only thanks to the intervention of the discrete group Γ_{u}.

(ii) Throughout this section, the Shimura datum (Q, X) can be replaced by (Q/U_{0}, U_{0}\X) for any rational subgroup U_{0} ⊂ U. In this way, the simplicial arguments of [HZ1,3.7] can be applied to extend the results of this section to the closed stratum Z_{P} of a toroidal compactification of (a connected component of) •Sh(G, X). This amounts to replacing Γ_{ℓ,P (P′)} by Γ_{ℓ,P} × Γ_{ℓ,P (P′)} on the righthand of formulas (1.5.27) and (1.5.33). As noted after (1.5.27), Γ_{ℓ,P} × Γ_{ℓ,P (P′)} is a congruence subgroup of G_{ℓ,R}(ℚ), and the results of this section come down to showing that two ways of calculating the coherent cohomology of Z_{P} give the same result.

(iii) The adelic formulations of Propositions 1.5.24 and 1.5.34 and Lemma 1.5.29 are left to the reader. These are intermediate results; the principal applications (in §3) will be stated in adelic form. In particular, the “long” calculation satisfies the analogue of Lemma 1.4.6.2.

2. Mixed growth conditions and coherent cohomology

In this Section, we treat the conditions of moderate growth and rapid decrease on differential forms—and also combinations of the two—along divisors with normal crossings and discuss their implications on Dolbeault cohomology. In the case of the boundary divisor of a toroidal compactification of the mixed Shimura variety associated to the pair (Q, X), we convert these conditions to growth and decay conditions on the corresponding vector-valued functions on the group Q on various Siegel sets for its admissible parabolic subgroups. This enables us to deduce the useful isomorphism (2.7.1) on cohomology that seems inaccessible by geometric methods.

(2.1) \overline{∂} with logarithmic growth in one variable (cf. [HP],[H4, §3], [HZ1, 3.8]). Let f be a C^{∞} function on the punctured disc Δ_{a}^{∗} = {z ∈ ℂ : 0 < |z| ≤ a}, with a < 1. Given an integer N, we say that f has logarithmic growth of degree ≤N when

(2.1.1)  |f(z)| |log |z||^{−N}

is bounded from above on Δ_{a}^{∗}. The set of all such functions is denoted A_{N}^{0}. It is obvious that A_{N}^{0} ⊂ A_{N′}^{0} for all N′ ≥ N.

One says that f is slowly increasing if it has logarithmic growth of degree ≤ N for some N, and rapidly decreasing if it has logarithmic growth of degree ≤ N for all N.^{5} It is wise to restrict one’s attention to functions f such that the derivatives of f of all orders are slowly increasing (resp. rapidly decreasing). The space of all such functions is denoted A_{sia}^{0} (resp. A_{rda}^{0}).

If ω is a differential form of bidegree (0,1), one can write ω = g(z)d\overline{z}/\overline{z}. We say that ω has logarithmic growth of degree ≤N whenever g does; in effect, we declare that d\overline{z}/\overline{z} is slowly increasing. We can then define the notion of (0,1)-forms that are slowly increasing or rapidly decreasing to all orders, and denote them by A_{sia}^{(0,1)} and A_{rda}^{(0,1)} respectively. We obtain Dolbeault complexes A_{sia}^{•}:

A_{sia}^{0} ⟶^{\overline{∂}} A_{sia}^{(0,1)},

and the analogously defined A_{rda}^{•}, which is a subcomplex of A_{sia}^{•}.

Proposition (2.1.2). (Dolbeault lemma with logarithmic growth). i) H^{0}(A_{sia}^{•}) is the space of holomorphic functions on the disc Δ_{a} of radius a.

ii) H^{0}(A_{rda}^{•}) is the space of holomorphic functions on Δ_{a} that vanish at the origin.

iii) H^{1}(A_{rda}^{•}) = H^{1}(A_{sia}^{•}) = 0.

As the first two assertions of the Proposition are obvious, we concentrate on (iii). This is seen to come down to verifying:

Proposition (2.1.3) [HP]. Let N be an integer different from −1. If g ∈ A_{N}^{0}, there exists f ∈ A_{N+1}^{0} such that \overline{∂}f(z) = g(z)d\overline{z}/\overline{z}.

For completeness, we give:

Outline of the proof of (2.1.3). There is a standard solution of the \overline{∂}-equation in a disc Δ, which gets written here as \overline{∂}f/∂\overline{z} = g(z)/\overline{z}, viz. f = I(g) where

(2.1.4)  I(g)(z) = (2πi)^{−1} ∫_{w∈Δ} \frac{g(w)dw ∧ d\overline{w}/\overline{w}}{w − z} .

This is an absolutely convergent integral if g has logarithmic growth; note that I(g) is independent of N. One then shows that logarithmic growth for g implies the corresponding growth for f. By decomposing (2.1.4) into pieces, as in [HP], one reduces the verification to the following elementary calculations:

5Clearly, one can assume without loss of generality that N is taken from any set of numbers that is unbounded from both above and below.

Lemma (2.1.5). Let N be an integer different from −1. Then there is a constant C_{N} > 0 such that whenever r ≤ a:

(i)  r^{−1} ∫_{0}^{r} |log ρ|^{N} dρ ≤ C_{N} |log r|^{N} ,

(ii+)  ∫_{0}^{r} ρ^{−1} |log ρ|^{N} dρ ≤ C_{N} |log r|^{N+1} , if N ≥ 0

(ii–)  ∫_{r}^{a} ρ^{−1} |log ρ|^{N} dρ ≤ C_{N} |log r|^{N+1} if N < 0.

Moreover, when g ∈ A_{sia}^{0} (resp. g ∈ A_{rda}^{0}), the formula

\frac{∂I(g)}{∂z} = I(\frac{∂g}{∂z})

implies that I(g) ∈ A_{sia}^{0} (resp. I(g) ∈ A_{rda}^{0}) likewise.

(2.2) Logarithmic growth in several variables. In the our setting, the natural notions of logarithmic growth and decay are the following. Given an integer N, a function f of (z, w) ∈ (Δ_{a}^{∗})^{m} × (Δ_{a})^{n} is said to have logarithmic growth of degree ≤N when

(2.2.1)  |f(z, w)| (∑_{j=1}^{m} |log |z_{j}||)^{−N}

is bounded from above. The set of all such functions is again denoted A_{N}^{0}. We can then define A_{sia}^{0} and A_{rda}^{0} as in (2.1). Also as in (2.1), we regard d\overline{z}_{j}/\overline{z}_{j} as logarithmic (after all, it has logarithmic growth in the sense of (1.3.5)). It is not hard to see that these growth conditions allow one to define, on any complex manifold M, sheaves of forms having slow increase or rapid decrease along the divisor with normal crossings D, for the notion of logarithmic growth is independent of local coordinates, and likewise for all derivatives. We denote these by A_{sia}^{•}(M, D) and A_{rda}^{•}(M, D).

An alternate simpler notion of logarithmic growth of degree N is provided by the boundedness of

(2.2.2)  |f(z, w)| ∏_{j=1}^{m} |log |z_{j}||^{−N} ,

which is asymptotically different from (2.2.1). However, (2.2.2) yields the same notions of slow increase or rapid decrease in view of the following:

Lemma (2.2.3). If λ_{j} ≥ 1 for 1 ≤ j ≤ m, then

(∏_{j=1}^{m} λ_{j})^{\frac{1}{m}} ≤ \frac{1}{m} ∑_{j=1}^{m} λ_{j} ≤ ∏_{j=1}^{m} λ_{j}.

This allow us to replace the growth condition of (2.2.1) by that of (2.2.2), for which we have, in effect, separation of variables (cf. [GH, p. 25]). One obtains from (2.2.3):

Proposition (2.2.4). Let D be a divisor with normal crossings on a complex manifold M. Then A_{sia}^{•}(M, D) is a fine resolution of O_{M}, and A_{rda}^{•}(M, D) is a fine resolution of O_{M}(−D).

Corollary (2.2.5). Let F be a locally-free sheaf of O_{M}-modules. Then A_{sia}^{•}(M, D)⊗_{O_{M}} F is a fine resolution of F, and A_{rda}^{•}(M, D) ⊗_{O_{M}} F is a fine resolution of F(−D).

(2.2.6) Remark. The issues involved in the proof of (2.2.4) were misrepresented in [HP], as was pointed out to us by J. I. Burgos, who also provided a correct argument along the lines outlined here.

(2.3) Forms with mixed growth conditions. We continue to let D denote a divisor with normal crossings on the complex manifold M. Consider any 2-partition of the set of irreducible components of D, and write D correspondingly as the union D_{1} ⋃ D_{2}. We define the middle term in

(2.3.1)  A_{rda}^{•}(M, D) ⊆ A_{mxa}^{•}(M; D, D_{2}) ⊆ A_{sia}^{•}(M, D)

by the condition that a differential form should be slowly increasing along D_{1}, and rapidly decreasing along D_{2}, and likewise for all derivatives. Our purpose is to obtain the following variants of (2.2.4) and (2.2.5):

Proposition (2.3.2). A_{mxa}^{•}(M; D, D_{2}) is a fine resolution of O_{M}(−D_{2}).

Corollary (2.3.3). For any locally-free sheaf F of O_{M}-modules, A_{mxa}^{•}(M; D, D_{2})⊗_{O_{M}} F is a fine resolution of F(−D_{2}).

One is led to define:

(2.3.4) Definition. A function f of (z, u, w) ∈ (Δ_{a}^{∗})^{m_{1}} × (Δ_{a}^{∗})^{m_{2}} × (Δ_{a})^{n} is said to have logarithmic growth of bidegree (N_{1}, N_{2}) when

|f(z, u, w)| (∑_{j=1}^{m_{1}} |log |z_{j}||)^{−N_{1}} (∑_{j=1}^{m_{2}} |log |u_{j}||)^{−N_{2}} .

is bounded from above. Differential (0, k)-forms of logarithmic growth of bidegree (N_{1}, N_{2}) are defined just as before, and the space of such is denoted A_{(N_{1},N_{2})}^{k}. Then Proposition (2.3.2) and indeed Proposition (2.2.4) are deduced from the following multivariate version of (2.1.3), which is proved by iteration on the latter.

Proposition (2.3.5). Let N_{1} and N_{2} be integers different from −1. If k > 0 and ω ∈ A_{(N_{1},N_{2})}^{k}, there exists ϕ ∈ A_{(N_{1}+1,N_{2}+1)}^{k−1} such that \overline{∂}ϕ = ω.

(2.4) Siegel sets in mixed Shimura varieties. We will retain the notation from (1.1) as far as possible. Thus, let (Q, X) be a mixed Shimura datum. By convention (as before) we also allow X to denote a connected component of this homogeneous space. Let (G, W \X) be the corresponding pure Shimura datum for G = Q/W and π : Q → G the natural map. For any connected component M_{Γ} of _{K_{f}} Sh(Q, X)(ℂ) and corresponding component M̂_{Γ} of K̂_{f} Sh(G, W \X)(ℂ), where K̂_{f} = π(K_{f}), we have the commutative diagram of complex manifolds

(2.4.1)  \begin{aligned}& X & ⟶ & M_{Γ} \\ & ↓ & & ↓ \\ X̂ = & W \X & ⟶ & M̂_{Γ} \end{aligned}

in which the spaces X and X̂ are contractible.

The notion of going to infinity in M_{Γ} or M̂_{Γ} is expressible in terms of Siegel sets, equivalently in the group or on the homogeneous space, associated to parabolic subgroups. For the present purposes, it is the admissible parabolic subgroups that enter. Recall from (1.1.9) that the mapping P(G) → P(Q) that assigns P̃ = π^{−1}(P) to P ∈ P(G) is bijective. Since the notion of admissibility is determined on G, the admissible parabolic subgroups of G and Q are thus in canonical one-to-one correspondence.

We work with the pair (G, X̂), though the story for (Q, X) is parallel. Let P ∈ P(G); then P (ℝ) acts transitively on X̂. Select a basepoint x̂_{0} ∈ X̂. Let A_{P} denote the associated lift to P (ℝ) of the connected component of the center of P/W_{P}. Then P^{0} ≃ (^{0}P)^{0} × A_{P}, where ^{0}P denotes, as in [BS, 1.1], the intersection of the kernels of the squares of the rational characters of P; we make this identification routinely. A Siegel set for P in G(ℝ) is a subset of P (ℝ) of the form 𝔖 = 𝔖_{κ,t} = κ × A_{P,t}, where κ is a compact subset of ^{0}P (ℝ), and

(2.4.2)  A_{P,t} = {a ∈ A_{P} : a^{β} ≥ t for all simple roots β occurring in W_{P} }.

It is also standard usage to call 𝔖 · x̂_{0} a Siegel set for P in X̂.

In what follows, U_{P} is the center of the unipotent radical W_{P} of P, V_{P} = W_{P} /U_{P}. The associated Siegel domain coordinates are deduced from embedding X̂ as an open subset of the total space X̂_{P} of a mixed Shimura variety,

(2.4.3)  X̂ ⊂ X̂_{P} → U_{P} (ℂ)\X̂_{P} → F̂_{P} ,

where F̂_{P} is the boundary component of X̂ corresponding to P, and trivializing the fibration analytically:

(2.4.4)  X̂_{P} ≃ F̂_{P} × V_{P} (ℝ) × U_{P} (ℂ).

This yields

(2.4.5)  X̂ ≃ {(y, v, u) ∈ F̂_{P} × V_{P} (ℝ) × U_{P} (ℂ) : Im u − Ψ(y, v) ∈ C_{P} },

where C_{P} is a homogeneous cone in U_{P} (ℝ), in which the action of A_{P} on X̂ is given by the dilations of C_{P}. In the above, Ψ : F̂_{P} × V_{P} (ℝ) → U_{P} (ℝ) has well-known properties; aside from continuity, these will not concern us here.

(2.4.6) Remark. When G is ℚ-simple and P is maximal, one places a point at infinity in V_{P} (ℝ) × U_{P} (ℂ) in (2.4.2) to attach (topologically) the Baily-Borel boundary component corresponding to P. It is reached by letting a^{β} → ∞ (see (2.4.2)), while keeping the simple roots of G_{h,P} bounded.

Next, let T be any compact cone in C_{P}; by “compact”, we mean that T̂ = (T − 0)/ℝ^{+} is compact in Ĉ_{P} := (C_{P} − 0)/ℝ^{+}. A complex Siegel set for P in X̂, built from T , will be a set S of the form

(2.4.7)  {(y, v, u) ∈ F̂_{P} × V_{P} × U_{P} (ℂ) : y ∈ κ_{h}, v ∈ κ_{V} , Re u ∈ κ_{U} , Im u ∈ (t_{0} + T )},

for κ_{h}, κ_{V}, κ_{U} compact and t_{0} ∈ T . It is easy to verify that the two notions of Siegel set are commensurate: every Siegel set for P is contained in a complex Siegel set for P, and conversely. (For the same reason, one sees that in defining a Siegel set, one could take the κ to be a compact subset of P (ℝ) or even of G(ℝ).)

We turn now briefly to the pair (Q, X). As our current concern is over mixed Shimura varieties for which π_{2} in (1.1.5) is an isomorphism (that is to say, abelian fibrations), we assume that W_{−2}Q = {1}. Let P̃ ∈ P(Q) correspond to P ∈ P(G).

There are objects for P̃ analogous to those for P given above, as defined in [P, §4]. This yields a diagram:

(2.4.8)  \begin{aligned}X & ⊂ & XP̃ & ⟶ & UP̃(ℂ)\XP̃ & ⟶ & FP̃ \\ ↓ & & ↓ & & ↓ & & ↓ \\ X̂ & ⊂ & X̂_{P} & ⟶ & U_{P} (ℂ)\X̂_{P} & ⟶ & F̂_{P} \end{aligned}

in which the vertical arrows are surjections, and a surjection of cones CP̃ ↠ C_{P}. (Note, however, that the homomorphism UP̃ → U_{P} need not be injective.) Moreover, if we select a basepoint x_{0} ∈ X with π(x_{0}) = x̂_{0}, the projection of AP̃ in Q(ℝ) to A_{P} ⊂ G(ℝ) is an isomorphism. Thus Siegel sets for P̃ in X, defined in the same way as those for P, map onto Siegel sets for P in X̂. In actuality, though, it will suffice for our purposes to consider the Siegel sets for P and P̃ independently.

The following is well-known (see [B3]):

Proposition (2.4.9). Let κ be a compact subset of ^{0}P (ℝ), 𝔖_{κ,t} the Siegel set κ × A_{P,t} and (𝔖_{κ,t})_{Γ} the image of 𝔖_{κ,t} in M̂_{Γ} ⊂ K̂_{f} Sh(G, X̂)(ℂ). Then for t sufficiently large,

(𝔖_{κ,t})_{Γ} ≃ κ_{Γ} × A_{P,t},

where κ_{Γ} is the image of κ in M̂_{Γ}, such that the projection 𝔖_{κ,t} → (𝔖_{κ,t})_{Γ} decomposes as (κ → κ_{Γ}) × 1_{A_{P,t}}.

Corollary (2.4.10). Let (𝔖_{κ,t})_{Γ} be as in (2.4.9). Then for t sufficiently large, there is a canonical projection a : (𝔖_{κ,t})_{Γ} → A_{P} (A_{P} -coordinate).

(2.5) Toroidal compactifications and Siegel sets. For simplifying notation only, we assume henceforth that G is ℚ-simple. We leave it to the reader to formulate the general case.

We start by taking a slight variation of the basic diagram (induced by (2.4.3)) in [HZ2, 1.4.4]:

(2.5.1)  \begin{aligned}M̃′_{P} = & Γ_{U_{P}} \X_{P} & ⟶ & M′_{P} \\ & ↓ & & ↓^{π_{2}} \\ & U_{P} (ℂ)\X_{P} & ⟶ & A_{P} \\ & ↓ & & ↓^{π_{1}} \\ & X̂_{P} & ⟶ & M_{P} \end{aligned}

(In [HZ1] A_{P} was denoted A_{F} or A_{P}, but here A_{P} has a different meaning.) We have made the upper square cartesian: M̃′_{P} is the pullback of M′_{P} to the universal cover U_{P} (ℂ)\X_{P} of A_{P}, replacing X_{P}. The coordinates in (2.4.4) induce an analytic isomorphism

(2.5.2)  M̃′_{P} ≃ X̂_{P} × V_{P} (ℝ) × T_{P} ,

where T_{P} denotes U_{P} (ℤ)\U_{P} (ℂ), viewed as the complex points of the torus with cocharacter group U_{P} (ℤ).

We next review some pertinent features of the structure of the boundary of toroidal compactifications M_{Γ,Σ} of connected components of Sh(G, X)(ℂ), following [HZ1, (1.5)]. Let P ∈ P(G). The data in Σ_{P} are used to construct π : (M′_{P})_{Σ_{P}} → M_{P}, with boundary divisor written as

(2.5.3)  ^{<}Z̃_{Σ}(⪰P) =^{def.} ⋃_{P′⪰P} ^{<}Z̃_{Σ}(P′).

The quotient of (2.5.3) by Γ_{ℓ,P} will be denoted ^{<}Z_{Σ}(⪰P); in the notation of [HZ1] it is the union of all ^{<}Z_{Σ}(P′) for P′ ⪰ P. Recall that a neighborhood of the latter is a building block in the construction of M_{Γ,Σ}. It will be useful to keep in mind that ^{<}Z_{Σ}(⪰P) is more than the P -stratum of the boundary unless P is maximal with respect to ≻, yet contains all of Z_{Σ}(P) (the closure of the P -stratum) only when P is minimal with respect to ≻. We also observe that T_{P} acts on (2.5.3).

The toroidal constructions with T_{P} lift to ones on M̃′_{P}. In particular, we have

(2.5.4)  (M̃′_{P})_{Σ_{P}} ≃ X̂_{P} × V × (T_{P})_{Σ_{P}} .

Let^{<} ˜Z̃_{Σ}(P) and^{<} ˜Z̃_{Σ}(⪰ P) denote the respective pullbacks of^{<}Z̃_{Σ}(P) and^{<}Z̃_{Σ}(⪰P); we can write, for instance,

(2.5.5)  ^{<} ˜Z̃_{Σ}(⪰P) ≃ X̂_{P} × V × ∂(T_{P})_{Σ_{P}} .

In [HZ2, 1.4(d)], we introduced a topological compactification of M_{Γ}, the real boundary quotient M_{Γ,Σ,∂_{ℝ}} of M_{Γ,Σ}, which was designed for the present discussion (also, cf. [HZ1, 2.3]), with structure map

p : M_{Γ,Σ} ⟶ M_{Γ,Σ,∂_{ℝ}}.

By construction, p(^{<}Z_{Σ}(⪰P)) is given naturally as Γ_{ℓ,P} \ (^{<}Z̃_{Σ}(⪰P)/T_{P}^{c} ), where T_{P}^{c} is the maximal compact real torus in T_{P}. Likewise, we can define the real boundary quotient

(2.5.6)  (M̃′_{P})_{Σ_{P} ,∂_{ℝ}} ≃ X̂_{P} × V × ∂(T_{P})_{Σ_{P} ,∂_{ℝ}},

for which ∂(M̃′_{P})_{Σ_{P} ,∂_{ℝ}} ≃ X̂_{P} × V × (T_{P})_{Σ_{P} ,∂_{ℝ}}. In terms of the product structures in (2.5.4) and (2.5.6), the quotient map (M̃′_{P})_{Σ_{P}} → (M̃′_{P})_{Σ_{P} ,∂_{ℝ}} is induced by the natural projection (T_{P})_{Σ_{P}} → (T_{P})_{Σ_{P} ,∂_{ℝ}}.

For any character χ of T_{P}, there is a morphism of commutative diagrams,^{6} which we also denote by χ:

(2.5.7)

\begin{cases}U_{P} (ℂ) & ⟶^{Im} & U_{P} (ℂ)/U_{P} (ℝ) ≃ U_{P} (ℝ) \\ ^{exp}↓ & & ^{exp}↓^{≃} \\ U_{P} (ℂ)/U_{P} (ℤ) ≃ T_{P} & ⟶_{Mod} & T_{P} /T_{P}^{c} \end{cases}\} ⟶^{χ} \begin{cases}ℂ & ⟶^{Im} & ℂ/ℝ ≃ ℝ \\ ^{exp}↓ & & ^{exp}↓^{≃} \\ ℂ/ℤ ≃ ℂ^{∗} & ⟶_{Mod} & ℂ^{∗}/S^{1} ≃ ℝ^{+} \end{cases}\}

In other words, up to a constant multiple, the logarithm of the modulus of a toroidal variable is induced by the imaginary part of a linear function on U_{P} (ℂ). In this way, approaching the boundary of X̃_{Σ,∂_{ℝ}} corresponds to going to infinity in the radial direction of the cone C_{P}. Specifically, ∂(T_{P})_{Σ_{P} ,∂_{ℝ}} can be viewed as a copy of Σ̂_{P} attached at infinity to the cone, with Σ̂_{P}^{(1),c} giving p(^{<}Z_{Σ}(P)) (see the illustration [HZ1, (2.3.10)]), while T_{P}^{c} -orbits are getting collapsed to a point.

We make the following: Observations (2.5.8). Let P be a maximal ℚ-parabolic subgroup of G, T a compact cone in C_{P}, and S a complex Siegel set for P built from T , as in (2.4.7). Then:

i) the closure of the image of S in (M̃′_{P})_{Σ_{P} ,∂_{ℝ}} meets the boundary in κ_{h} ×κ_{V} ×T , ii) Take T to be any union of finitely many top-dimensional simplicial cones in

Σ_{P}^{(1),c} that maps onto Γ_{ℓ,P} \Σ_{P}^{(1),c}, and assume that κ_{V} and κ_{U} are sufficiently large. Then the closure of the image of S in M_{Γ,Σ} meets the boundary in π^{−1}(κ̂_{h}) ∩^{<}Z_{P}, where κ̂_{h} denotes the image of κ_{h} in M_{P}.

iii) For any P′ ≻ P, there is a dense open subset (determined by Y_{P} (P′) below) of Z_{Σ}(P′) of which each point is in the closure of some Siegel set for P. In practice, the above asserts that every point of Z_{Σ}(P) (the union of components of the toroidal divisor generated by Σ_{P}^{c}) is a limit point of some Siegel set; however, one must consider Siegel sets for all parabolic subgroups dominated by (i.e., ≺) P, as well as those for P itself.

To be more specific, if we let V (L) stand for the set of vertices of a simplicial complex L, then

(2.5.9)  V (Σ̂_{P}) = ∐{V (Σ̂_{P′}^{c}) : P′ ⪰P }

6In the literature (see [HZ1, (2.3.2)]), the induced map T_{P} → U_{P} (ℝ) is called “ord”.

is a Γ_{ℓ,P} -equivariant partition (cf. [HZ2, (2.4.2)]). By taking the real projection of the toroidal divisors corresponding to each Σ̂_{P′}^{c}, one induces a decomposition of Ĉ_{P} (which is an open subset of |Σ̂_{P} |) into sets Ŷ_{P} (P′) that intersect only at their boundaries, and which are the closure of their interiors; of these, only Ŷ_{P} (P) is compact modulo Γ_{ℓ,P}. In [HZ2, (2.2)], Ŷ_{P} (P′) gets called the open thick P′-stratum of Σ̂_{P}^{(1)}, and the decomposition (2.5.9) plays an important role. When T ⊂ C_{P} as before, we will say that T abuts on Z̃_{Σ}(P′) whenever T̂ ⊂ Ŷ_{P} (P′). Letting Y_{P} (P′) denote the corresponding cone in C_{P}, we conclude:

Proposition (2.5.10). Let P be a maximal ℚ-parabolic subgroup of G. Then:

i) The set of all complex Siegel sets for admissible parabolics that abut on Z̃_{Σ}(P) consists precisely of those complex Siegel sets, for some Q ⪯ P, built from compact cones contained in the subcone Y_{Q}(P) of C_{Q}.

ii) Let Z̃_{Σ}^{Q}(P) denote the union of the limit points of such Siegel sets for a fixed Q. Then this set is open and dense in Z̃_{Σ}(P).

iii) Let Z_{Σ}^{Q}(P) ⊆ Z_{Σ}(P) denote the image of Z̃_{Σ}^{Q}(P) in M_{Γ,Σ}. Then

Z_{Σ}(P) = ⋃{Z_{Σ}^{Q}(P) : Q ∈ M(Γ)},

where M(Γ) is a set a Γ-conjugacy classes of admissible parabolic subgroups Q with Q ⪯ P.

iv) ⋂{Z_{Σ}^{Q}(P) : Q ∈ M(Γ)} is an open and dense subset of Z_{Σ}(P). It is open in the Zariski topology only when P is minimal with respect to ⪯, i.e., when M(Γ) has only one element (representable by {P }).

(2.6) Growth conditions at the toroidal boundary. Given P ∈ P(G), let β denote the positive simple root occurring in W_{P}. We now prove:

Proposition (2.6.1). Let (z, w) ∈ (Δ_{r}^{∗})^{n} × Δ_{r}^{m}, with r < 1, give local coordinates on a deleted neighborhood of ^{<}Z_{Σ}(⪰P) associated to an n-fold intersection of components of this divisor. Then on the intersection of this coordinate neighborhood with (𝔖_{κ,t})_{Γ}, the functions a^{β} (determined by (2.4.10)) and ∑_{j=1}^{n} log |z_{j}| grow at the same rate along ^{<}Z_{Σ}(⪰P).

An argument proving this is given for pure Shimura varieties in [Mu, p. 264], though it seems a bit unfocused. The proof is geometric, and goes the same in the mixed case; we present it here:

Proof. We appeal to (2.5.7). Write λ_{j} for log |z_{j}|; these are linear functions on the cone that are positive in the region under consideration. We may assume without loss of generality that the Siegel set (𝔖_{κ,t})_{Γ} involves, as a factor, a compact subset of the “standard” cross-section of C_{P} −{0} → Ĉ_{P} determined by G_{ℓ,P} (ℝ)^{der}. Then, the value of a^{β} at any point is given by the amount of dilation in C_{P} required to get to the point from this cross-section. By compactness, this differs from the amount of dilation required to get from any other cross-section over the same compact set by an amount that is bounded from above and away from zero. In other words, the rate of growth of the A_{P} -coordinate is independent of cross-section. Then, just take the cross-section to be the one defined by ∑_{j} λ_{j} = 1.

Next, let x̂_{0} be the basepoint of X̂, which was used in definition of Siegel sets in (2.4). Let K̂_{0} denote the stabilizer in G(ℝ)^{der} of x̂_{0}. It is well-known (see [B1]) that there are norms || · || on G(ℝ)^{der} that satisfy:

i) ||gk|| = ||g|| whenever k ∈ K̂_{0};

ii) ||gg′|| ≤ ||g||||g′||. The following is easy to verify: Proposition (2.6.2). Let P be a maximal parabolic subgroup of G. Then:

i) On A_{P}, ||a|| grows like (a^{β})^{ℓ} for some ℓ > 0;

ii) For g in the Siegel set 𝔖_{κ,t}, ||g|| and ||a(g)|| have the same rate of growth. We recall that a C^{∞} (0, q)-form ω on M_{Γ} with values in W is determined by a

unique smooth Γ-invariant function f_{ω} on G(ℝ) with values in ⋀^{q}(𝔭^{−})^{∗} ⊗W. When we combine the preceding with (2.5.10), we get: Proposition (2.6.3).

i) An element ω of the Dolbeault complex for W on (an open subset of) M_{Γ} has moderate growth along Z_{Σ}(P) in the sense of (2.2.4) if and only if f_{ω} has moderate growth on all complex Siegel sets for Q that are built from subcones of Y_{Q}(P) ⊂ C_{Q}, for all Q ⪯ P.

ii) ω lies in A_{mxa}^{•}(M_{Γ,Σ}; Z_{Σ}, Z_{Σ}(P)) ⊗ W^{can} (of (2.3)) if and only if f_{ω} has moderate growth and has rapid decrease on all complex Siegel sets for Q that are built from subcones of Y_{Q}(P) ⊂ C_{Q}, for all Q ⪯ P.

(2.7) Mixed growth conditions and basechange. We now take in (2.4.1) the lower part of the tower for the mixed Shimura variety for the admissible parabolic subgroup P of G (this is denoted π_{1} : A_{P} → M_{P} in (2.5.1)). According to [P], there exist compatible toroidal compactifications for π_{1}. Let π̃_{1} : Ã_{P,Ξ} → M̃_{P,Ξ} (as in [HZ1, 1.6]) be one of them. We let Z_{Ξ}^{A}(P′) and Z_{Ξ}(P′) resp. denote the P′-strata of the respective toroidal boundaries. The main goal of this section is to deduce the following variant and extension of [HZ1, 3.14]:

Proposition (2.7.1). In the above situation, there is a commutative diagram of quasi-isomorphisms of sheaves on M̃_{P,Ξ}:

Rπ̃_{1,∗}{[V^{A}]^{can}(−π_{1}^{∗}Z_{Ξ}(P′))} ⟶^{≈} Rπ̃_{1,∗}{[V^{A}]^{can}}(−Z_{Ξ}(P′))
↓ ↓^{≈}
Rπ̃_{1,∗}{[V^{A}]^{can}(−Z_{Ξ}^{A}(P′))} ↔^{≈} H^{•}(𝔰_{P} , V)^{can}(−Z_{Ξ}(P′)) = ⨁_{i} H^{i}(𝔰_{P} , V)^{can}(−Z_{Ξ}(P′))[−i]

This implies at once:

Corollary (2.7.2). In the above situation,

H^{•}(Ã_{P,Ξ}, [V^{A}]^{can}(−Z_{Ξ}^{A}(P′))) ≃ H^{•}(M̃_{P,Ξ}, H^{•}(𝔰_{P} , V)^{can}(−Z_{Ξ}(P′))).

(2.7.3) Remark. There would be nothing to prove in (2.7.1) if one knew that Z_{Ξ}^{A}(P′) were equal to π_{1}^{∗}Z_{Ξ}(P′) in the sense of divisors, i.e., that the fibers of π̃_{1} over Z_{P′} are reduced. One might hope that this can be arranged by choosing Ξ suitably, but that is not apparent from [P]. Proof of (2.7.1). We show first that in the derived category of M̃_{P,Ξ}, both sides of the isomorphism in the bottom row of (2.7.1), are computable by means of complexes of C^{∞} differential forms, on M_{P} and CalA_{P} respectively, with moderate growth and that decrease rapidly in the direction of the P′-strata.

We use the mechanism from [HZ1, 3.5.12]. Consider the Dolbeault complexes:

(2.7.4)  A_{M_{P}}^{0,•} (H^{•}(𝔰_{P} , V)) ↪^{ι_{1}} A_{M_{P}}^{0,•} (⋀ 𝔰_{P}^{∗} ⊗ V) ↪^{ι_{2}} π_{1}_{∗}A_{A_{P}}^{0,•} (V^{A});

up to quasi-isomorphism, this is

(2.7.5)  H^{•}(𝔰_{P} , V) → ⋀ 𝔰_{P}^{∗} ⊗ V → Rπ_{1,∗}V^{A}.

The complexes in (2.7.4) can be written as sheaves of relative Lie algebra cochains:

(2.7.6)  C_{1}^{•} ↪ C_{2}^{•} ↪ C_{3}^{•},

where, for O open in M_{P},

(2.7.6.1)  C_{1}^{•}(O) = {C^{∞}(O_{h,P}) ⊗ ⋀(𝔭_{h}^{−})^{∗} ⊗ H^{•}(𝔰_{P} , V)}^{K_{h,P}} ,

(2.7.6.2)  C_{2}^{•}(O) = {C^{∞}(O_{h,P}) ⊗ ⋀(𝔭_{h}^{−})^{∗} ⊗ ⋀(𝔰_{P})^{∗} ⊗ V }^{K_{h,P}} ,

where O_{h,P} is the inverse image of O under Γ_{h,P} \G_{h,P} → M_{P}; also for O′ the inverse image of O in A_{P}

(2.7.6.3)  C_{3}^{•}(O) = {C^{∞}(O′_{1}) ⊗ ⋀(𝔭_{h}^{−} ⊕ 𝔰_{P})^{∗} ⊗ V }^{K_{h,P}} ,

with O′_{1} the inverse image of O_{h,P} in Γ_{A_{P}} \(P′/U_{P})(ℝ). For (2.7.6.3), we are using the fact that V^{A} is a homogeneous vector bundle (recall (1.2.14)), determined by the representation of K_{h,P} on H^{0}(T_{P} , V′_{P})^{T_{P}}, which is isomorphic (via evaluation at the basepoint) to V as a K_{h,P} -module.

There are homotopy inverses to the maps in (2.7.6) via projections

(2.7.7)  C_{1}^{•} ⟵^{ψ_{1}} C_{2}^{•} ⟵^{ψ_{2}} C_{3}^{•},

with ψ_{1} given by the semi-simplicity of representations of a compact group—note that we are invoking here the assumption that 𝔭^{−} acts trivially on V —and ψ_{2} given by taking the constant term along the fibers of A_{P}.

Here and again, one makes use of the following homological lemma:

Lemma (2.7.8). Let C_{1} and C_{2} be cochain complexes [of sheaves], and assume given morphisms i : C_{1} → C_{2} and ψ : C_{2} → C_{1} such that

i) ψ ∘ i = 1,

ii) i ∘ ψ = 1 + dB + Bd. for some mapping B : C_{2} → C_{1} of degree −1. Let C′_{2} = ker ψ. If i′ : C′_{2} → C_{2} denotes the inclusion, the morphism

C_{1} ⊕ C′_{2} ⟶^{i ⊕i′} C_{2}.

is an isomorphism. Moreover, C′_{2} ≈ 0, and i and ψ are quasi-isomorphisms inducing mutually inverse isomorphisms on cohomology [sheaves]. Furthermore, the above is functorial with respect to the triple (i, ψ, B).

This yields that the maps in (2.7.4), and therefore also (2.7.5), are quasi-isomorphisms. This was extended to the toroidal boundary for the canonical extension in [HZ1, 3.13] and the subcanonical extension in [HZ1, 3.14]. These were obtained by observing that ψ_{1} and ψ_{2} respect the growth conditions on (2.7.7) given as in (2.6.1), as do their homotopy operators, and then appealing to (2.7.8). Said another way, imposing the growth conditions preserved the homological situation, and we could again invoke (2.7.8).

We apply (2.3.3) to both sides of bottom row in (2.7.1), writing them as Dolbeault complexes with mixed growth conditions on Ã_{P,Ξ} and M̃_{P,Ξ} resp., taking D_{2} to be the respective P′-stratum of the toroidal boundary, for some ℚ-parabolic subgroup Q of P. This reduces us to verifying that (2.7.9) A_{mxa}^{•}(M̃_{P,Ξ}; Z_{Ξ}, Z_{Ξ}(P′))⊗H^{•}(𝔰_{P} , V)^{can} → π̃_{1,∗}{A_{mxa}^{•}(Ã_{P,Ξ}; Z_{Ξ}^{A}, Z_{Ξ}^{A}(P′))⊗[V^{A}]^{can}}.

is a quasi-isomorphism, extending that of (2.7.4) to the toroidal boundary. For use in conjunction with the Lie algebra cochains of (2.7.6), we appeal to (2.6.3, ii): the mixed growth conditions are equivalent to moderate growth, together with decay conditions on a specific class of Siegel sets. That these are preserved by ψ_{1} and ψ_{2} in (2.7.7) follows as before. This gives that (2.7.9) is a quasi-isomorphism, and (2.7.1) follows.

To complete the proof of (2.7.1), we must show that the square commutes. First, we make the following observation, which follows from (2.2.5) and (2.3.3):

Lemma (2.7.10). The inclusions

(2.7.10.1) A_{sia}^{•}(Ã_{P,Ξ}, Z_{Ξ}^{A})⊗[V^{A}]^{can}⊗π^{∗}I_{Z_{Ξ}(P′)} ↪ A_{mxa}^{•}(Ã_{P,Ξ}; Z_{Ξ}^{A}, Z_{Ξ}^{A}(P′))⊗[V^{A}]^{can} ↪ A_{sia}^{•}(Ã_{P,Ξ}, Z_{Ξ}^{A})⊗[V^{A}]^{can}

are compatible fine resolutions of

(2.7.10.2)  [V^{A}]^{can} ⊗ π^{∗}I_{Z_{Ξ}(P′)} ↪ [V^{A}]^{can} ⊗ I_{Z_{Ξ}^{A}(P′)} ↪ [V^{A}]^{can}.

Applying the functor Rπ̃_{∗} to (2.7.10.1) gives

π̃_{∗}{A_{sia}^{•}(Ã_{P,Ξ}, Z_{Ξ}^{A}) ⊗ [V^{A}]^{can}} ⊗ I_{Z_{Ξ}(P′)} ↪ {π̃_{∗}A_{mxa}^{•}(Ã_{P,Ξ}; Z_{Ξ}^{A}, Z_{Ξ}^{A}(P′)) ⊗ [V^{A}]^{can}}
↪ {π̃_{∗}A_{sia}^{•}(Ã_{P,Ξ}, Z_{Ξ}^{A}) ⊗ [V^{A}]^{can}}.

The above are taken as C_{2}’s in (2.7.8); for C_{1}’s, we take

A_{sia}^{•}(M̃_{P,Ξ}, Z_{Ξ}) ⊗ H^{•}(𝔰_{P} , V)^{can} ⊗ I_{Z_{Ξ}(P′)} ↪ A_{mxa}^{•}(M̃_{P,Ξ}, Z_{Ξ}) ⊗ H^{•}(𝔰_{P} , V)^{can}
↪ A_{sia}^{•}(M̃_{P,Ξ}, Z_{Ξ}) ⊗ H^{•}(𝔰_{P} , V)^{can}.

From here, it is obvious that the square in (2.7.1) commutes.

The above implies a useful fact:

Proposition (2.7.11). The morphism

H^{•}(Ã_{P,Ξ}, [V^{A}]^{can}) → H^{•}(Ã_{P,Ξ}, [V^{A}]^{can} ⊗ O_{Z_{Ξ}^{A}(P′)})

is given by the direct sum of mappings

H^{•}(M̃_{P,Ξ}, H^{i}(𝔰_{P} , V)^{can}) → H^{•}(M̃_{P,Ξ}, H^{i}(𝔰_{P} , V)^{can} ⊗ O_{Z_{Ξ}(P′)}).

Proof. The sheaf [V^{A}]^{can} ⊗ O_{Z_{Ξ}^{A}(P′)} is quasi-isomorphic to the complex

(2.7.11.1)  {[V^{A}]^{can} ⊗ I_{Z_{Ξ}^{A}(P′)} → [V^{A}]^{can}}[1]

Then

Rπ̃_{∗}{[V^{A}]^{can} ⊗ I_{Z_{Ξ}^{A}(P′)} → [V^{A}]^{can}}[1]

is quasi-isomorphic to

(2.7.11.2)  ⨁_{i}{H^{i}(𝔰_{P} , V)^{can} ⊗ I_{Z_{Ξ}(P′)} → H^{i}(𝔰_{P} , V)^{can}}[1],

which is in turn quasi-isomorphic to H^{•}(𝔰_{P} , V)^{can} ⊗O_{Z_{Ξ}(P′)}. Restriction to Z_{Ξ}^{A}(P′) and Z_{Ξ}(P′) is induced by inclusion into (2.7.11.1) and (2.7.11.2). Our assertion follows.

3. The nerve spectral sequence for coherent cohomology

For the remainder of the article, we fix a pure (standard) Shimura datum (G, X). Thus G is a reductive group over ℚ; for simplicity we assume G^{ad} to be ℚ-simple. All mixed Shimura varieties will be realized as boundary strata of toroidal compactifications of _{K_{f}} Sh(G, X)_{Σ}. Here and in what follows the notation K_{f} and Σ are used as in §1, relative to (G, X), and will be used without comment. Moreover, all toroidal data Σ will be assumed to be full, in the sense of [HZ1, 2.2.6].

(3.1) Cohomology on boundary strata of toroidal compactifications. We refer to (0.5)–(0.7) and (1.1.9) for notation and conventions concerning rational parabolics and boundary strata. For R ∈ P(G), R subordinate to P, we define the R-stratum Z_{Σ}(R) of _{K_{f}} Sh(G, X)_{Σ} as in (0.6.1) and [HZ2, (1.5.2)]. We revert momentarily to the classical (non-adelic) language of [HZ1, §§1–3]. When R = P is maximal, Z_{Σ}(R) is itself the union of irreducible divisors with normal crossings corresponding to 1-dimensional cones in Σ_{P}^{c} modulo conjugation by Γ_{ℓ,P} (notation as in [HZ1], esp. §1.5). The nerve 𝔑_{Σ}(R) of the closed covering of Z_{Σ}(R) by its irreducible components (denoted 𝔑(Z_{Σ_{P}}) in [HZ1]) is a simplicial complex canonically isomorphic to Γ_{ℓ,P} \Σ̂_{P}^{c} [HZ1, Lemma 3.7.2]. More generally, if r is the parabolic rank of R, then Z_{Σ}(R) has a closed covering by irreducible components of codimension r in K_{f} Sh(G, X)_{Σ}. However, the nerve 𝔑_{Σ}(R) of this closed covering does not have a transparent description in terms of Σ_{F}. This is because, for instance, if P ≺ P_{1}, R = P ∩ P_{1}, and σ (resp. σ′) ∈ Σ_{P} is a two-dimensional (resp. three-dimensional) cone with one (resp. two) edge(s) in Σ_{P_{1}} and one in Σ_{P}^{c}, then the corresponding closed toroidal stratum \overline{Z}_{σ} (resp. \overline{Z}_{σ′}) is of codimension 0 (resp. 1) in Z_{Σ}(R) but the facets of σ and σ′ in Σ_{P} have the same dimension. The constructions in [HZ2, §2] were designed to deal with this problem.

Let i_{R} : Z_{Σ}(R) ↪ _{K_{f}} Sh(G, X)_{Σ} denote the canonical closed immersion. For R = P maximal, the geometric description of 𝔑_{Σ}(P) in [HZ1, §3] yields a simple expression for the coherent cohomology H^{•}(Z_{Σ}(P), i_{P}^{∗} [W]^{can}), when [W] is an automorphic vector bundle on Sh(G, X) [HZ1, 3.7.8, 3.13.6]. The main step is the calculation [HZ1, 3.6.4] in terms of the pure Shimura variety attached to G_{h,P} of the cohomology H^{•}(\overline{Z}_{σ}, i_{σ}^{∗}[W]^{can}) where \overline{Z}_{σ} is an individual closed stratum as above and i_{σ} : \overline{Z}_{σ} ↪ _{K_{f}} Sh(G, X)_{Σ} is the inclusion. More precisely, the argument in [HZ1, 3.6.4] only concerns the open stratum Z_{σ}, and expresses its cohomology in terms of that of a family of automorphic vector bundles V_{λ(h,w)} on the base M_{P}. For \overline{Z}_{σ} we obtain the analogous expression in terms of the canonical extensions [V_{λ(h,w)}]^{can} on the toroidal compactification M_{P,Σ(P)}, using Proposition 1.4.4 and Corollary 1.4.5; see (3.1.2), below. In particular, the calculation works in exactly the same way when 𝔑_{Σ}(P) is replaced by 𝔑_{Σ}(R) for general R.

Remark. Careful readers of the proofs in [HZ1] may note an apparent dependence on analytic considerations not discussed here. Specifically, the reduction of the cohomology of a boundary stratum to cohomology of reductive groups is carried out in Corollary 3.7.8 of [HZ1]. An intermediate step in the proof of this corollary is provided by Lemma 3.7.5 of [op. cit.], which (in the present setting) calculates the coherent cohomology of Z_{Σ}(P) in terms of the Γ_{ℓ,P} -equivariant coherent cohomology of an abelian scheme, the analogue of Sh(Q/U, U\X)_{Σ_{2}}. In turn, the proof of Lemma 3.7.5 makes forward reference to [HZ, 3.9.4], whose proof is analytic and homotopy-theoretic. However, that argument was not needed; Lemma 3.7.5 is actually a simple consequence of (3.7.3), (3.7.4), and (3.7.6).

We first state the result (for general R) in the classical setting. Thus D is a connected component of X and Γ ⊂ G is a neat arithmetic subgroup of G(ℝ) that fixes X. We retain the notation Γ_{ℓ,R} = Γ ∩ G_{ℓ,P} (ℚ) ∩ R(ℚ), when R is subordinate to P, from [HZ2]. We fix a point p ∈ D as in [HZ1, §3.5], let K_{p} ⊂ G(ℝ) denote its stabilizer. We can and do always choose p so that K_{p} ∩ P (ℝ) contains a maximal connected compact subgroup of P (ℝ) for every standard parabolic P. As in [HZ1], we restrict attention to fully decomposed automorphic vector bundles; i.e., those associated to representations of K_{p} ([HZ1, Definition 3.1.2]).

We let M_{P} denote the (connected) pure Shimura variety associated to P. Similarly, we let W^{P,p} denote the subset of the Weyl group of K_{p} from [HZ1, §3.6] (see (0.5)). Recall that this is the set of Kostant representatives for the maximal parabolic denoted Q^{P,p} ⊂ K_{p} with Levi subgroup K_{p}^{(2)} and unipotent radical S_{p}. The Lie algebras are denoted by lower case Gothic characters, as usual. Recall further [HZ1, (1.8.3), (1.8.6)] that the Cayley transform induces isomorphisms

(3.1.0)  K_{p}^{P,(2)} ⟶^{∼} K_{h,P} × G_{ℓ,P}; 𝔰_{p,P} ⟶^{∼}𝔳_{p}^{−}.

Here K_{h,P} ⊂ G_{h,P} is the maximal compact subgroup associated to p. More generally, if R is subordinate to the maximal parabolic P, we let Q^{R,p} ⊂ Q^{P,p} be the parabolic with Levi factor isomorphic via Cayley transform to K_{h,P} × G_{ℓ,R} and unipotent radical S_{p,R} ⊃ S_{p,P} and let W^{R,p} be the corresponding set of Kostant representatives in the Weyl group of K_{p}.

Proposition (3.1.1). Let [W] be an irreducible automorphic vector bundle on Sh(G, X) and let R be a proper ℚ-parabolic subgroup of G, subordinate to the maximal parabolic P. Then

(i) For each b ≥ 0, the assignment

(3.1.1.1)  σ ↦ H^{b}(\overline{Z}_{σ}, i_{σ}^{∗}[W]^{can})

defines a locally constant sheaf L^{b}(·, W) on the simplicial complex 𝔑_{Σ}(R), and an associated spectral sequence

(3.1.1.2)  E_{1}^{a,b} = H^{a}(𝔑_{Σ}(R), L^{b}(·, W)) ⇒ H^{a+b}(Z_{Σ}(R), i_{R}^{∗}[W]^{can}).

(ii) Over the universal cover of 𝔑_{Σ}(R), L^{b}(•, W) is isomorphic to

(3.1.1.3)  ⨁_{w∈W^{P,p}} H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ V_{λ(ℓ,w)}.

Here M_{P,Σ(P)} is any admissible toroidal compactification of M_{P}, ℓ(w) is the length of w and V_{λ(h,w)} and V_{λ(ℓ,w)} are defined (relative to [W]) as in [HZ1, (3.6.1)].

(iii) The spectral sequence in (i) degenerates at E_{1}, and is naturally split.

Remarks (3.1.2). i) The hypothesis that [W] be irreducible is made in order to obtain the simple decomposition (3.1.1.3). This hypothesis was inadvertently omitted from the discussion in [HZ1, §3].

ii) For R = P maximal, the above is a strengthening of (3.7.3), (3.7.4) and (3.7.7) of [HZ1], which pertain to the (open) P -stratum. (Note the misprint in [loc. cit], where the spectral sequence was labelled E_{2} rather than E_{1}.) Here, we extend those results in two ways: we are working with the closed R-stratum Z_{Σ}(R), and this is for arbitrary R. Correspondingly, the above formulas involve canonical extension to some toroidal compactification M_{P,Σ(P)}.

Proof of (3.1.1). We adapt the line of reasoning in [HZ1,§3]. By [P, Corollary 7.17], every closed stratum \overline{Z}_{σ} ⊂ Z_{Σ}(R) (whose interior was constructed from a cone in Σ_{P}) can be realized as a toroidal compactification of a (connected) mixed Shimura variety of the form _{K_{f}} Sh(Q, X)_{Σ}, of the sort considered in (1.4). For appropriately chosen Σ, we get a morphism

π_{σ} = π_{2,σ} ∘ π_{1,σ} : \overline{Z}_{σ} → M_{P,Σ(P)}

as in (1.4.2). To obtain the formulas in (3.1.1), we incorporate growth conditions at the toroidal boundary, replacing Corollary 3.6.3 of [loc. cit.] by Proposition 1.4.4 here. Then, Corollary 1.4.5 (the generalization of the Leray spectral sequences of [HZ1, 3.5.10]) reads:

E_{2}^{p,q} = H^{p}(M_{P,Σ(P)}, (R^{q}Φ_{σ}i_{σ}^{∗}[W])^{can}) ⇒ H^{p+q}(\overline{Z}_{σ}, i_{σ}^{∗}[W]^{can}) = L^{p+q}(σ, W).

Moreover, as noted in Remark (1.2.16.3), i_{σ}^{∗}[W]^{can} satisfies a Kostant condition in this case. Hence this Leray spectral sequence degenerates at E_{2} and the grading induced by the E_{2}-term splits canonically:

\begin{aligned}L^{b}(σ, W) & = ⊕_{q}H^{b−q}(M_{P,Σ(P)}, (R^{q}Φ_{σ}i_{σ}^{∗}[W])^{can}) \\ & = ⨁_{w∈W^{P,p}} H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ V_{λ(ℓ,w)}. \end{aligned}

The second equality, the decomposition of the higher direct image as a sum over Kostant representatives, is (the version with growth conditions of) [HZ1,Corollary 3.6.4].

Note that the final expression is independent of σ. As in [HZ1, §3.6], this expression is compatible with all restriction maps for the simplicial complex 𝔑_{Σ}(R) [loc. cit., 3.6.4] and with the action of the fundamental group of 𝔑_{Σ}(R) ([loc. cit, 3.6.3]; see below). This proves (i) and (ii), and implies that the E_{1}-term in (3.1.1.2) is canonically isomorphic to

⨁_{w∈W^{P,p}} H^{a}(π_{1}(𝔑_{Σ}(R)), V_{λ(ℓ,w)}) ⊗ H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}).

(The fundamental group of 𝔑_{Σ}(R) is identified with Γ_{ℓ,R} in Proposition 3.1.3, below.) Let Ṽ_{λ(ℓ,w)} denote the local system on 𝔑_{Σ}(R) associated o the representation of the fundamental group on V_{λ(ℓ,w)}. Thus (iii) follows from the Künneth formula, or more precisely from the universal coefficient theorem, applied to the sum over w ∈ W^{P,p} of the local systems

Ṽ_{λ(ℓ,w)} ⊗ H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can})

on 𝔑_{Σ}(R), as in the last part of the proof of Proposition 3.7.7 of [HZ1].

In §2 of [HZ2] the nerves 𝔑_{Σ}(R) are identified for general R, up to homotopy type. Let X(Γ_{ℓ,P}) be the locally symmetric space associated to the arithmetic subgroup Γ_{ℓ,P} (denoted X(Γ_{ℓ}) in [HZ1, 2.2.10]). Let X(Γ_{ℓ,P}) ↪ \overline{X}(Γ_{ℓ,P}) denote the Borel-Serre compactification. The result is the following:

Proposition (3.1.3). Let R be a proper ℚ-parabolic subgroup of G, subordinate to the maximal parabolic P. As above, let R_{ℓ,P} denote the corresponding ℚ-parabolic subgroup of G_{ℓ,P}. Then

(i) The nerve 𝔑_{Σ}(R) is homotopy equivalent to the R_{ℓ,P} -stratum \overline{e′(R_{ℓ,P})} ⊂ \overline{X}(Γ_{ℓ,P}).

(ii) Let R ⊂ R′ be two ℚ-parabolic subgroups of G subordinate to P. Under the homotopy equivalences in (i), the natural inclusion 𝔑_{Σ}(R) ⊂ 𝔑_{Σ}(R′) corresponds to the natural inclusion \overline{e′(R_{ℓ,P})} ⊂ \overline{e′(R′_{ℓ,P})}.

This is essentially contained in Proposition 2.6.4 of [HZ2]; the translation into the present language is provided by Corollaries 2.5.9 and 2.5.10 of [loc. cit.].

The fundamental group of \overline{e′(R_{ℓ,P})} is just Γ_{ℓ,R}. Let Ṽ_{λ(ℓ,w)} denote the local system on \overline{X}(Γ_{ℓ,P}) associated to the representation λ(ℓ, w), as in the proof of (3.1.1) (cf. the statement of Corollary 3.7.8 of [HZ1]). Just as in the case of maximal strata (cf. [HZ1, Corollary 3.7.8]), Proposition 3.1.3 and (3.1.2) allow us to identify

H^{a}(𝔑_{Σ}(R), L^{b}(•, W)) ≃ ⊕_{w∈W^{P,p}} H^{a}(Γ_{ℓ,R}, H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ V_{λ(ℓ,w)})
≃ ⊕_{w∈W^{P,p}} H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ H^{a}(Γ_{ℓ,R}, V_{λ(ℓ,w)})

≃ ⊕_{w∈W^{P,p}} H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ H^{a}(\overline{e′(R_{ℓ,P})}, i_{R_{ℓ,P}}^{∗} Ṽ_{λ(ℓ,w)}) (3.1.4)

where the second isomorphism expresses the fact that Γ_{ℓ,R} acts trivially on the first factor, and in the third isomorphism i_{R_{ℓ,P}} : \overline{e′(R_{ℓ,P})} ↪ \overline{X}(Γ_{ℓ,P}) is the natural inclusion.

On the other hand, the calculation of H^{•}(\overline{e′(R_{ℓ,P})}, i_{R_{ℓ,P}}^{∗} Ṽ_{λ(ℓ,w)}) is standard. Let W_{ℓ,P} denote the relative Weyl group (over ℚ) of G_{ℓ,P}, and let W_{ℓ}^{R} ⊂ W_{ℓ,P} denote the subset of Kostant representatives relative to R_{ℓ,P} (cf. Notation). Identify G_{ℓ,R} with a Levi quotient of R_{ℓ,P} and let Γ_{ℓ,R}^{red} denote the image of Γ_{ℓ,R} in G_{ℓ,R}(ℚ); let X(Γ_{ℓ,R}^{red}) be the corresponding locally symmetric space, the quotient by Γ_{ℓ,R}^{red} of the symmetric space attached to G_{ℓ,R}. Let μ(ℓ, w) denote the highest weight of λ(ℓ, w) relative to the choice of positive roots for G_{ℓ,P} made in [HZ1, 3.6]. Then [Ha]

(3.1.5)  H^{a}(\overline{e′(R_{ℓ,P})}, i_{R_{ℓ,P}}^{∗} Ṽ_{λ(ℓ,w)})⟶^{∼} ⨁_{ω∈W_{ℓ}^{R}} H^{a−ℓ(ω)}(X(Γ_{ℓ,R}^{red}), Ṽ_{λ(ℓ,w;ω)})

Here Ṽ_{λ(ℓ,w;ω)} is the local system on X(Γ_{ℓ,R}^{red}) associated to the representation λ(ℓ, w; ω) of G_{ℓ,R}, and this in turn is the representation with highest weight ω(μ(ℓ, w)+ ρ_{ℓ}) − ρ_{ℓ}, where ρ_{ℓ} is the half-sum of positive roots of G_{ℓ,P}.

The following lemma will be used repeatedly in what follows.

Lemma (3.1.6). Let H be a reductive group over the field k and let P_{1} ⊃ P_{2} be a pair of k-parabolic subgroups of G, with Levi decompositions P_{1} = L_{1} · U_{1}, P_{2} = L_{2} · U_{2}, L_{1} ⊃ L_{2}. Let Q = P_{2} ∩ L_{1}. Let T ⊂ L_{2} be a maximal k-split torus and let W = W (H, T) and W_{1} = W (L_{1}, T) be the relative Weyl groups. Choose a minimal parabolic subgroup P_{0} ⊂ P_{2}. Let W^{i} ⊂ W be the sets of Kostant representatives for P_{i}, i = 1, 2; let W^{Q} ⊂ W_{1} be the set of Kostant representatives for Q. (In both cases the Kostant representatives are defined relative to the ordering determined by P_{0}.) Under the natural identification of W_{1} with a subgroup of W, we then have

W^{2} = {ω · w | ω ∈ W^{Q}, w ∈ W^{1}}.

Moreover, with the above notation, ℓ_{W} (ω · w) = ℓ_{W} (w) + ℓ_{W_{1}}(ω), where ℓ_{W} and ℓ_{W_{1}} are the length functions on W and W_{1}, respectively.

Proof.

Applying Lemma 3.1.6, with P_{1} = Q^{P,p}, P_{2} = Q^{R,p}, and thus Q ≃ R_{ℓ,P}, to (3.1.4) and (3.1.5), we thus find for (3.1.1,(i))

E_{2}^{a,b} ≃ ⨁_{w=w_{1}·w_{2}∈W^{R,p}} H^{b−ℓ(w_{2})}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can})⊗H^{a−ℓ(w_{1})}(X(Γ_{ℓ,R}^{red}), Ṽ_{λ(ℓ,w)}).

Here w ∈ W^{R,p} is factored as in Lemma 3.1.6 as w_{1} · w_{2} with w_{1} ∈ W_{ℓ}^{R} and w_{2} ∈ W^{P,p}; ℓ(w_{i}) is given by the appropriate length function; and λ(ℓ, w) = λ(ℓ, w_{2}; w_{1}) in the previous notation. Moreover, we have used Kostant’s theorem to write

(3.1.7)  H^{i}(𝔰_{p,R}, V_{λ}) = ⨁_{w∈W^{R,p};ℓ(w)=i} V_{λ(h,w)} ⊗ V_{λ(ℓ,w)}

where V_{λ(h,w)} = V_{λ(h,w_{2})} is a representation of K_{h,P} and V_{λ(ℓ,w)} is a representation of G_{ℓ,R}.

Combining this with (3.1.1,(iii)), we obtain the following formula:

Proposition (3.1.8). Let [W] be an irreducible automorphic vector bundle on Sh(G, X), and let R be a proper ℚ-parabolic subgroup of G, subordinate to the maximal parabolic P. Then

H^{k}(Z_{Σ}(R), i_{R}^{∗}[W]^{can}) ≃ ⨁_{a+b=k} ⨁_{w∈W^{R,p}} H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can})⊗H^{a}(X(Γ_{ℓ,R}^{red}), Ṽ_{λ(ℓ,w)}).

(3.1.9) Remark. (i) In the setting of Lemma 3.1.6, let 𝔲_{i} denote the Lie algebra of U_{i} for i = 1, 2 and 𝔲_{Q} the Lie algebra of the unipotent radical of Q. Then 𝔲_{2} = 𝔲_{1} ⊕ 𝔲_{Q}. If follows from Lemma (3.1.6) that the spectral sequence

E_{2}^{p,q} = H^{q}(𝔲_{Q}, H^{p}(𝔲_{1}, V)) ⇒ H^{p+q}(𝔲_{2}, V).

degenerates at E_{2}.

(ii) Here and in what follows, the passage from the sum over W^{P,p} to a sum over W^{R,p} should be regarded as a purely topological operation. All the arithmetic information is carried by the G_{h,P} -factor, and is independent of the parabolic subgroup R subordinate to P. In particular:

(a) The isomorphism (3.1.5) is rational over the field of coefficients of the automorphic vector bundle [W]. This is because the spectral sequence in (i) above, with Q = R_{ℓ,P}, is purely algebraic and 𝔲_{Q} is a ℚ-rational nilpotent Lie subalgebra of 𝔤. In particular, the adelic reformulation (3.2.9) of the calculations of this section is compatible with the canonical models of Sh(G, X) and Sh(G_{h,P} , X(P)).

(b) Similarly, for the Hodge-theoretic applications in §4, it makes no difference whether the cohomology of the R-stratum is written as a sum indexed by W^{P,p} or by W^{R,p}. (Compare (4.2.27) and (4.2.29)). Again, this is because the contribution of the G_{ℓ,R}-factor is purely topological.

(3.2) The nerve spectral sequence for boundary cohomology. The spectral sequence in question is the one associated to the closed covering {Z_{Σ}(P) : P maximal} of Z_{Σ}. In classical language, the E_{1}-term is

(3.2.1)  E_{1}^{r,s} = ⨁_{r(R)=r+1} E_{1}^{r,s}(R) ⇒ H^{r+s}(Z_{Σ}, i^{∗}[W]^{can}), where

(3.2.2)  E_{1}^{r,s}(R) = H^{s}(Z_{Σ}(R), i_{R}^{∗}[W]^{can}).

Using the description of the right-hand side of (3.2.2) given in (3.1.8), we can write the above as (3.2.3)

E_{1}^{r,s}(R) = ⨁_{a+b=s} ⨁_{w∈W^{R,p}} H^{b−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ H^{a}(X(Γ_{ℓ,R}^{red}), Ṽ_{λ(ℓ,w)}).

For R of parabolic rank r + 1 and R′ ⊂ R of parabolic corank 1, we will see that the nature of the contribution to d_{1},

d_{1}(R, R′) : E_{1}^{r,s}(R) → E_{1}^{r+1,s}(R′),

depends on whether or not Π(R′) = Π(R) (in the notation of (0.7)).

We set up the adelic version of the above. We drop the level subgroup K_{f} from the notation and write Sh_{Σ} for Sh(G, X)_{Σ}, ∂Sh_{Σ} for its (toroidal) boundary, Sh_{Σ}^{R(∗)} for its R(∗)-stratum, and \overline{Sh}_{Σ}^{R(∗)} for the Zariski closure of Sh_{Σ}^{R(∗)} in ∂Sh_{Σ}. As in [HZ1, 5.3], the \overline{Sh}_{Σ}^{P (∗)}, as P runs through the maximal admissible parabolics, form a closed cover of ∂Sh_{Σ}. Taking the nerve of this closed cover thus yields a spectral sequence for the cohomology of ∂Sh_{Σ} with coefficients in a canonically extended automorphic vector bundle. For simplicity we describe this spectral sequence when G^{ad} is assumed ℚ-simple. Let i_{R} : \overline{Sh}_{Σ}^{R(∗)} ↪ Sh_{Σ} denote the corresponding closed embedding, and let r(R) denote the parabolic rank of R. Taking the limit over neat open compact subgroups and families of fans, we obtain the nerve spectral sequence in the following form:

(3.2.4)  E_{1}^{r,s} = ⨁_{r(R)=r+1} \underrightarrow{lim}_{K_{f} ,Σ} H^{s}(\overline{Sh}_{Σ}^{R(∗)}, i_{R}^{∗}([W]^{can})) ⇒ H^{r+s}([W](∞)).

Here H^{•}([W](∞)) denotes \underrightarrow{lim}_{K_{f} ,Σ} H^{•}(∂Sh_{Σ}, [W]^{can} ⊗ O_{∂Sh_{Σ}}), as in [HZ1].

The individual summands on the left-hand side of (3.2.4) can be written as induced representations, just as in [HZ1]. It is most convenient to express each term as a tensor product of the coherent cohomology of a (pure) Shimura variety by the cohomology of a locally symmetric space. Let R be any rational parabolic, subordinate to the maximal rational parabolic P, and define G_{ℓ,R} as before. We define the adelic locally symmetric space

(3.2.5)  X(G_{ℓ,R}) = \underleftarrow{lim} G_{ℓ,R}(ℚ)\G_{ℓ,R}(A)/(K_{ℓ,R} · Z_{ℓ,R})(ℝ) · K_{ℓ,R,f},

where Z_{ℓ,R} is the center of G_{ℓ,R}, K_{ℓ,R} = K_{p} ∩ G_{ℓ,R}(ℝ), and K_{ℓ,R,f} runs through compact open subgroups of G_{ℓ,R}(A_{f}). Let L_{R} be a (standard) Levi subgroup of R. We also let Sh(G_{h,P} , X(P)) denote the Shimura variety attached to G_{h,P}. For brevity we write Sh(R) = Sh(G_{h,P} , X(P)) when R is subordinate to P. For each w ∈ W^{R,p} we have the automorphic vector bundle [V_{λ(h,w)}] over Sh(R), as above, and therefore we can define the cohomology of its canonical extensions in the adelic limit

H̃^{•}([V_{λ(h,w)}])

(notation as in [H4] and [HZ1, §4]). We define

(3.2.6)  H^{•}(w) = H̃^{•}([V_{λ(h,w)}]) ⊗ H^{•}(X(G_{ℓ,R}), Ṽ_{λ(ℓ,w)})[−ℓ(w)].

by analogy with [HZ1, (4.1.10)]

Parametrization of connected components introduces annoying complications, as in [HZ1, 4.1.12]. We let

Δ_{1,R} = Ker : G_{h}(A_{f}) × G_{ℓ,R}(A_{f}) → G_{h}(A_{f}) · G_{ℓ,R}(A_{f})

(the product viewed as a subgroup of R(A_{f})) and let

(3.2.7)  Δ_{0,R} = L_{R}(ℚ)^{+}/(G_{h}(ℚ)^{+} · G_{ℓ,R}(ℚ)^{+}); Δ_{R} = Δ_{0,R} × Δ_{1,R}.

(For the relation between (3.2.7) and connected components of the R stratum when R is maximal, see [HZ1, (4.1.10,4.1.11)]. The analogous relation holds for general R.) Then G_{h}(A_{f}) × G_{ℓ,R}(A_{f}) acts on H^{•}(w), and therefore so does its subgroup Δ_{1,R}. Moreover Δ_{0,R} acts on Sh(R) × X(G_{ℓ,R}) by the analogue of [HZ1, (4.1.11)]. We define

(3.2.8)  I^{R}{H^{•}(w)} = Ind_{G_{h}(A_{f})·G_{ℓ,R}(A_{f})⋊Δ_{0,R}}^{L_{R}(A_{f})}{H^{•}(w)^{Δ_{1,R}}},

just as in [HZ1, (4.1.13)]. Then we have Corollary (3.2.9). i) In the spectral sequence (3.2.4), there is a natural decomposition E_{1}^{r,s} = ⨁_{r(R)=r+1} E_{1}^{r,s}(R), where

E_{1}^{r,s}(R) = Ind_{R(A_{f})}^{G(A_{f})} ⨁_{i} ⨁_{w∈W^{R,p}} I^{R}{H̃^{s−i−ℓ(w)}([V_{λ(h,w)}]) ⊗ H^{i}(X(G_{ℓ,R}), Ṽ_{λ(ℓ,w)})}.

ii) Moreover, suppose R′ ⊂ R are two ℚ-parabolic subgroups of G, both subordinate to P, with r(R′) = r + 2, r(R) = r + 1. Let Q′ = R′_{ℓ,P}, Q(R′) = Q′ ∩ G_{ℓ,R}. Let d_{R′,R} denote the restriction E_{1}^{r,s}(R) of the nerve spectral sequence differential d_{1} : E_{1}^{r,s} → E_{1}^{r+1,s}, followed by projection to E_{1}^{r+1,s}(R′). Then, with respect to the above formula, d_{R′,R} is induced by the identity on the holomorphic component tensored with the boundary map induced by the inclusion of the boundary stratum \overline{e′(Q(R′))} in the Borel-Serre compactification of X(G_{ℓ,R}).

Proof. The first statement generalizes Corollary 4.1.14 of [HZ1], and is proved in the same way. The second statement follows immediately from (ii) of Proposition 3.1.3 (as such, it deduced from the same for (3.2.2)).

(3.3) Differentials in the nerve spectral sequence. Corollary 3.2.9 completes the verification of [HZ1, (5.3.12.4)], and computes half of the differentials d_{1} in the nerve spectral sequence in terms of the cohomology of locally symmetric spaces attached to reductive groups. The present section will begin the computation of the remaining half: the d_{R′,R} for r(R′) = r(R) + 1 with R and R′ subordinate to distinct maximal parabolics, say P and P′, respectively. Let P (P′) = P′ ∩ G_{h,P}, as in (1.4.6). In terms of the expression for E_{1}^{r,s} in Corollary 3.2.9, d_{R′,R} will be expressed in terms of the restriction map from the first (holomorphic) tensor factor to the boundary stratum corresponding to P (P′). In other words, d_{R′,R}, which is given a priori by a restriction map on the coherent cohomology of mixed Shimura varieties – more precisely, the cohomology of simplicial schemes whose components are all toroidal compactifications of mixed Shimura varieties – can be expressed as a restriction map whose source is the coherent cohomology of a toroidally compactified pure Shimura variety. This is the opposite of the story in (ii) in Corollary 3.2.9, and will be proved in (3.4) below.

As usual, we will verify this at the level of connected Shimura varieties at finite level, the extension to the adelic setting being routine. Thus we have isomorphisms (3.3.1)

E_{1}^{r,s}(R)⟶^{∼} ⊕_{w∈W^{R,p}} ⊕_{i}H^{s−i−ℓ(w)}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ H^{i}(X(Γ_{ℓ,R}^{red}), Ṽ_{λ(ℓ,w)})

and (3.3.2)

E_{1}^{r+1,s}(R′)⟶^{∼} ⨁_{w′∈W^{R′,p}} ⊕_{j}H^{s−j−ℓ(w′)}(M_{P′,Σ(P′)}, [V_{λ(h,w′)}]^{can})⊗H^{j}(X(Γ_{ℓ,R′}^{red}), Ṽ_{λ(ℓ,w′)})

On the other hand, letting [B] denote any automorphic vector bundle on M_{P,Σ(P)}, we have by (3.1.8) applied to G_{h,P}

(3.3.3) H^{t}(\overline{Sh_{Σ}^{P (P′)(∗)}}, i_{P (P′)}^{∗}[B]^{can})⟶^{∼}

⨁_{ω∈W^{P (P′),p′}} H^{t−k−ℓ(ω)}(M_{P′,Σ(P′)}, [B_{λ(h,ω)}]^{can}) ⊗ H^{k}(X(Γ_{ℓ,P (P′)}), B̃_{λ(ℓ,ω)})

Here [B_{λ(h,ω})] and B̃_{λ(ℓ,ω)} bear the same relation to [B] (for the boundary component M_{P′,Σ(P′)} of M_{P,Σ(P)}) as [V_{λ(h,ω})] and V_{λ(ℓ,ω)} bear to [W] (for the boundary component M_{P,Σ(P)} of Sh(G, X)). Finally, p′ is the fixed point of the subgroup K_{p} ∩ G_{h,P}. Bear in mind that

(3.3.4)  G_{ℓ,R′} = G_{ℓ,R} · G_{ℓ,P (P′)},

where G_{ℓ,P (P′)} ⊂ G_{h,P} is the G_{ℓ}-factor of the maximal parabolic P (P′) ⊂ G_{h,P}. Then for sufficiently small Γ, we have

Γ_{ℓ,R′} = Γ_{ℓ,R} × Γ_{ℓ,P (P′)}; Γ_{ℓ,R′}^{red} = Γ_{ℓ,R}^{red} × Γ_{ℓ,P (P′)}.

Moreover, Lemma 3.1.6 allows us to write

W^{R′,p} = W^{P (P′),p′} · W^{R,p}.

Say w′ ∈ W^{R′,p} is the product ω · w, with ω ∈ W^{P (P′),p′} and w ∈ W^{R,p}. We then have canonical isomorphisms

(3.3.5)  V_{λ(ℓ,w′)}⟶^{∼}V_{λ(h,w)(ℓ,ω)} ⊗ V_{λ(ℓ,w)};

(3.3.6)

H^{j}(X(Γ_{ℓ,R′}^{red}), Ṽ_{λ(ℓ,w′)})⟶^{∼}⊕_{i+k=j}H^{k}(X(Γ_{ℓ,P (P′)}), Ṽ_{λ(h,w)(ℓ,ω)})⊗H^{i}(X(Γ_{ℓ,R}^{red}, Ṽ_{λ(ℓ,w)}).

Here V_{λ(h,w)(ℓ,ω)} is the representation of G_{ℓ,P (P′)} associated to the highest weight λ(h, w) of K_{h,P} and the Kostant representative ω.

We consider the toroidally compactified (connected) Shimura variety M_{P,Σ(P)}; let

i_{P (P′)} : Z_{Σ(P)}(P (P′)) ↪ M_{P,Σ(P)}

denote the inclusion of the closed P (P′) stratum of its boundary. If in (3.3.3) we now let [B] vary over the [V_{λ(h,w)}] from (3.3.1), we conclude that the right-hand side of (3.3.2) is the sum over w of the right-hand sides of (3.3.3), tensored with the sum over i of H^{i}(Γ_{ℓ,R}, V_{λ(ℓ,w)}): (3.3.7)

E_{1}^{r+1,s}(R′)⟶^{∼} ⨁_{w∈W^{R,p}} ⨁_{i} E_{1}^{0,s−i−ℓ(w)}(P (P′); [V_{λ(h,w)}]^{can}) ⊗ H^{i}(X(Γ_{ℓ,R}^{red}, Ṽ_{λ(ℓ,w)}).

Here we have written E_{1}^{0,s}(P (P′); [B]) for the P (P′) piece of the E_{1}^{0,s}-term of the nerve spectral sequence calculating the boundary cohomology of [B]^{can} on M_{P,Σ(P)}; it is the cohomology of the pullback of [V_{λ(h,w)}] to Z_{Σ(P)}(P (P′)).

For use in the next section, we note that the above factorization of W^{R′,p} can be continued. As in (3.1), we have Levi subgroups K_{p}^{P′,(2)} ⊂ K_{p}, K_{p′}^{P (P′),(2)}, etc., and the corresponding sets of Kostant representatives. Now we have the factorizations:

(3.3.8) \begin{aligned}W^{R′,p} & = W^{P (P′),p′} · W^{R,p} = W^{P (P′),p′} · W^{P,p} · W_{ℓ}^{R} = W^{P ∩P′,p} · W_{ℓ}^{R} \\ & = W^{Q_{ℓ}} · W^{P′,p} · W_{ℓ}^{R}. \end{aligned}

Here W_{ℓ}^{R} is a set of Kostant representatives for R ∩ G_{ℓ,P} in G_{ℓ,P} and the factorization W^{R,p} = W^{P,p} · W_{ℓ}^{R} is the application of Lemma 3.1.6, via Cayley transform, to the Levi subgroup K_{h,P} × G_{ℓ,R} ⊂ K_{h,P} · G_{ℓ,P}. Similarly, W^{P ∩P′,p} is a set of Kostant representatives in K_{p} obtained by identifying K_{p}^{P,(2)} ⟶^{∼} K_{h,P} × G_{ℓ,P} and then taking the parabolic subgroup associated to P (P′) in K_{h,P}; the last equality on the first line is the application of Lemma 3.1.6 to the pair of inclusions

K_{p′}^{P (P′),(2)} × G_{ℓ} ⊂ K_{h,P} × G_{ℓ}

Finally, we can identify K_{p′}^{P (P′),(2)} ⟶^{∼} K_{h,P (P′)} × G_{ℓ,P (P′)} via Cayley transform on G_{h,P}, and K_{h,P (P′)} is just K_{h,P′}. Applying Lemma 3.1.6 to the standard parabolic R′_{ℓ} of G_{ℓ,P′} with Levi factor G_{ℓ,R′} = G_{ℓ,R} × G_{ℓ,P (P′)}, we obtain the factorization W^{P ∩P′,p} = W^{R′_{ℓ}} · W^{P′,p}, whence the second line.

(3.4) Differentials in the nerve spectral sequence, concluded. For any automorphic vector bundle [B] on M_{P}, let

d_{P (P′)} : H^{•}(M_{P,Σ(P)}, [B]^{can})→H^{•}(Z_{Σ(P)}(P (P′)), i_{P (P′)}^{∗}[B]^{can})

denote the natural restriction. We use the same notation to designate the homomorphism

H^{•}(M_{P,Σ(P)}, [B]^{can})→E_{1}^{0,•}(P (P′), B)

obtained by composing the restriction map with the decomposition (3.2.3), and analogously in the adelic setting.

Proposition (3.4.1). Let [W] be an automorphic vector bundle on Sh(G, X), as above, and let R and R′ be a pair of proper parabolic subgroups of G, with R′ ⊂ R r(R′) = r(R) + 1. Suppose that R and R′ are subordinate to the maximal parabolics P and P′ respectively (P′ ≺ P). In terms of the (adelic versions of the) isomorphisms (3.3.1) and (3.3.7), the map d_{R,R′} of Corollary (3.2.9) is given by

⨁_{w∈W^{R,p}} ⨁_{i} d_{P (P′)} ⊗ 1_{i,λ(ℓ,w)},

where 1_{i,λ(ℓ,w)} is the identity map of H^{i}(X(G_{ℓ,R}), Ṽ_{λ(ℓ,w)}).

Proof. The proposition is stated in the adelic framework, but for the proof it suffices to work with connected Shimura varieties. Let \overline{Z}_{σ} ⊂ Z_{Σ}(R) be an irreducible stratum and let \overline{Z}_{σ(R′)} = \overline{Z}_{σ} ∩ Z_{Σ}(R′) = \overline{Z}_{σ} ∩ Z_{Σ}(P′). Let i_{σ} and i_{σ(R′)} be the inclusions of \overline{Z}_{σ} and \overline{Z}_{σ(R′)}, respectively, in _{K_{f}} Sh(G, X)_{Σ}.

We will show that the following diagram commutes: (3.4.2)

H^{•}(\overline{Z}_{σ}, i_{σ}^{∗}([W]^{can})) ⟶ H^{•}(\overline{Z}_{σ(R′)}, i_{σ(R′)}^{∗}([W]^{can}))
f↑  ^{g′}↑
⨁_{w∈W^{P}} H^{•}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⟶^{d_{P (P′)}} ⨁_{w∈W^{P}} H^{•}(Z_{Σ(P)}(P (P′)), i_{P (P′)}^{∗}[V_{λ(h,w)}]^{can})

Here the top arrow is the natural restriction map, the left-hand vertical arrow is given by Corollary 1.4.5, and the right-hand vertical arrow is to be constructed. There is difficulty caused by the fact that \overline{Z}_{σ(R′)}, by which we mean the reduced divisor on \overline{Z}_{σ}, need not be the pullback of Z_{Σ(P)}(P (P′)).

We abstract the situation and study the restriction map d_{R′,R} in the context of base change. Consider the following diagram of morphisms of schemes of finite type over ℂ:

(3.4.3)  \begin{aligned}Z′′ = Z′_{red} & ⟶^{i_{0}} & Z′ & ⟶^{i′} & A \\ & & ^{g}↓ & & ^{π}↓ \\ & & Z & ⟶^{i} & M \end{aligned}

We assume that the square is cartesian. The notation Z′_{red} refers to the underlying reduced scheme of Z′. The schemes A and M are assumed smooth, and Z is a (reduced) divisor on M. The morphism π is proper. More generally, we will be later considering situations in which (3.4.4) i) A is a finite union of irreducible components, each of which is a smooth scheme, as are the intersections of any finite set of irreducible components;

ii) any intersection of irreducible components of A is smooth and the restriction of π to it is proper. In effect, we will take A to be a simplicial scheme, each of whose components is as in (3.4.3).

Let F be a locally-free sheaf on A. Consider the following diagram:

(3.4.5)  \begin{aligned}0 & ⟶ & F ⊗ I_{Z′} & ⟶ & F & ⟶ & F ⊗ O_{Z′} & ⟶ & 0 \\ & & ↓ & ^{=}↓ & & ↓ \\ 0 & ⟶ & F ⊗ I_{Z′′} & ⟶ & F & ⟶ & F ⊗ O_{Z′′} & ⟶ & 0 \end{aligned}

The rows are exact because F is locally-free. Applying the (exact) functor Rπ_{∗} to the first row, we get

(3.4.6)  0 ⟶ Rπ_{∗}(F ⊗ I_{Z′}) ⟶ Rπ_{∗}F ⟶ Rπ_{∗}(i^{′∗}F) ⟶ 0.

Since I_{Z′} = π^{∗}I_{Z} and I_{Z} is locally-free, we can rewrite (3.4.6) by “adjunction” as

(3.4.7)  0 ⟶ (Rπ_{∗}F) ⊗ I_{Z} ⟶ Rπ_{∗}F ⟶ Rπ_{∗}(i^{′∗}F) ⟶ 0.

This yields the isomorphism

(3.4.8)  Rπ_{∗}(i^{′∗}F) ≈ i^{∗}Rπ_{∗}F.

We get from (3.4.5):

(3.4.9)  \begin{aligned}0 & ⟶ & S ⊗ I_{Z} & ⟶ & S & ⟶ & Rπ_{∗}(i^{′∗}F) & ⟶ & 0 \\ & & ↓ & ^{=}↓ & & ↓ \\ 0 & ⟶ & Rπ_{∗}(F ⊗ I_{Z′′}) & ⟶ & S & ⟶ & Rπ_{∗}(i^{′′∗}F) & ⟶ & 0 \end{aligned}

where we have written S for Rπ_{∗}F, and i′′ = i′ ∘ i_{0}.

We consider the diagram (3.4.3), where π : A → M is a morphism of toroidally compactified mixed Shimura varieties _{•}Sh(Q, X)_{Σ} ⟶ _{•}Sh(Q′, X′)_{Σ_{1}}, with Q′ = Q/R_{u}Q (so M is pure), and Z is the (reduced) boundary divisor associated to a maximal parabolic subgroup of Q′. Take F to be a canonically extended fully decomposed automorphic vector bundle [V^{A}]^{can} on A, and S = Rπ_{∗}F again. We want to compare the restriction map H^{•}(A, F) ⟶ H^{•}(Z′′, i^{′′,∗}F) to the map H^{•}(M, S) ⟶ H^{•}(Z, i^{∗}S).

We are actually interested in the case in (3.4.2): A = \overline{Z}_{σ}, M = M_{P,Σ(P)}, Z = Z_{Σ(P)}(P (P′)), and F = i_{σ}^{∗}([W]^{can}). However, we see by Proposition 1.4.3 that in general we may replace Q by Q/W_{−2}Q, thereby reducing the issue to the case where W_{−2}Q = (0). Then π : A → M above is generically an abelian scheme, and that puts us into the situation of Proposition 2.7.1 (with P′ replaced here by P (P′) and Z′′ = Z_{P′}^{A}).

In the present situation, it follows from Proposition 2.7.1 that the left vertical arrow in (3.4.9) is a quasi-isomorphism. This provides the top half of the following commutative diagram, where the vertical arrows are just the Leray spectral sequence: (3.4.10)

H^{•}(A, F)  ⟶  H^{•}(A, i^{′,∗}F)  ⟶^{≃}  H^{•}(A, i^{′′,∗}F)
≃↑  ^{≃}↑  ^{≃}↑
H^{•}(M, Rπ_{∗}F) ⟶ H^{•}(M, Rπ_{∗}(i^{′,∗}F)) ⟶^{≃} H^{•}(M, Rπ_{∗}(i^{′′,∗}F))
↑↓  ↑↓  ↑↓
H^{•}(M, ⊕ S_{w}) ⟶ H^{•}(M, ⊕ i^{∗}S_{w})  =  H^{•}(M, ⊕ i^{∗}S_{w})

As for the bottom half of (3.4.10), the pairs of opposing vertical maps (all isomorphisms) are given by the splitting in the derived category provided by (2.7.6) and (2.7.7) for all three instances. More precisely, from Lemma (2.7.10) and the subsequent discussion we obtain the left half of the following commutative diagram: (3.4.11)

H^{•}(A, F ⊗ I_{Z′})  ⟶  H^{•}(A, F)  ⟶  H^{•}(A, i^{′,∗}F)
≃↑  ^{≃}↑  ^{≃}↑
H^{•}(M, Rπ_{∗}F ⊗ I_{Z})) ⟶ H^{•}(M, Rπ_{∗}F) ⟶ H^{•}(M, Rπ_{∗}(i^{′,∗}F))
↑↓ ↑↓ ↑↓
H^{•}(M, ⊕ S_{w} ⊗ I_{Z}) ⟶ H^{•}(M, ⊕ S_{w}) ⟶ H^{•}(M, ⊕ i^{∗}S_{w})

The top of (3.4.11) is obtained from (3.4.7). Using (3.4.8), we can replace Rπ_{∗}(i^{′,∗}F) by i^{∗}Rπ_{∗}F in the second row of (3.4.11). That the last pair of opposing vertical maps are isomorphisms then follows by the five lemma. It follows that restriction in (3.4.10) respects the splittings. This completes the verification of (3.4.2).

Now the natural projection Z_{Σ(P)}(P (P′))→M_{P′,Σ(P′)} —we are assuming our toroidal compactifications have been chosen compatibly—defines isomorphisms

g′′ : H^{•}(Z_{Σ(P)}(P (P′)), i_{P (P′)}^{∗}[V_{λ(h,w)}]^{can})⟶^{∼}E_{1}^{0,•}(P (P′), [V_{λ(h,w)}]).

We claim that the composite

g′′ ∘ (g′)^{−1} : H^{•}(\overline{Z}_{σ(R′)}, i_{σ(R′)}^{∗}([W]^{can}))⟶^{∼} ⊕_{w∈W^{P}} E_{1}^{0,•}(P (P′), [V_{λ(h,w)}])

equals the isomorphism given by applying Proposition 1.4.5 simplicially to the individual closed strata of \overline{Z}_{σ(R′)}.

More precisely, if \overline{Z}_{σ′} is an irreducible stratum of \overline{Z}_{σ(R′)}, then (1.4.5), combined with (3.1.2), asserts an isomorphism between H^{•}(\overline{Z}_{σ′}, i_{σ′}^{∗}([W]^{can})) and a sum over terms indexed by W^{P′,p}. Similarly, (3.3.3) writes each E_{1}^{0,•}(P (P′), [V_{λ(h,w)}]) as a sum of terms indexed by W^{P (P′),p′}. As in (3.3.8), we have

W^{P (P′),p′} · W^{P,p} = W^{P ∩P′,p} = W^{R′_{ℓ}} · W^{P′,p}.

Thus this sum can be regrouped as a sum over terms indexed by W^{P′,p}. The claim, from which the proposition will follow, is that the composite g′′ ∘ (g′)^{−1} can be identified with the isomorphism of (1.4.5) after regrouping in this way. But this follows formally by applying Proposition 1.5.34 simplicially to the individual closed strata (cf. Remark 1.5.35 (ii)).

We conclude as in [HZ1, §4.8]:

Theorem (3.4.12). Let [W] be an automorphic vector bundle on Sh(G, X), with canonical model over the field E(W). Suppose R′ ⊂ R are two ℚ-parabolics of G with r(R′) = r +2 = r(R)+1. Write E_{1}^{r,s}(R) and E_{1}^{r+1,s}(R′) as in Corollary 3.2.9. Then the differential d_{R,R′} : E_{1}^{r,s}(R) → E_{1}^{r+1,s}(R′) is rational with respect to the E(W)-structures on the two sides for all s.

Proof. If R′ and R are subordinate to two distinct maximal parabolics, this follows from Theorem 4.8.1 of [HZ1] and Proposition 3.4.1, above. If R′ and R are subordinate to the same maximal parabolic, the theorem is a simple consequence of Corollary 3.2.9.

(3.5) Towards the determination of E_{2}. It is a good idea at this point to recall how the nerve spectral sequence (3.2.1) arises as the spectral sequence of a filtered complex. Writing C^{•}(·) for any functorial cochain complex representing RΓ(·), one has that C^{•}(i^{∗}[W]^{can}) is quasi-isomorphic to the double complex K^{•}, with terms

(3.5.1)  K^{r,s} = ⨁_{r(R) = r+1} C^{s}(i_{R}^{∗}[W]^{can}),

in which the differential d is the sum of the differential of C^{•} (which increases s by one), and restriction (which increases r by one). We denote the latter by d_{par}. The filtration R according to the value of r is a decreasing filtration of K^{•}, for which

(3.5.2)  Gr_{R}^{t}K^{•} = ⨁_{r(R) = t+1} C^{•}(i_{R}^{∗}[W]^{can})[−t],

in which d_{par} vanishes. This is the source of the nerve spectral sequence, with E_{1}-term E_{1}(K^{•}, R), and d_{1} induced by d_{par}.

For comparison later, it is convenient to replace parabolic rank by actual rank. Let ρ denote the rank of G (over ℚ). A ℚ-parabolic subgroup R of G has ℚ-rank ρ_{R}, with ρ_{R} +r(R) = ρ. Thus, r(R) = t+1 if and only if ρ_{R} = ρ−1−t. The increasing filtration by ℚ-rank determines the same spectral sequence (up to a translation) that R does; we allow R to refer to that from now on.

Our calculations in this Section suggest the following notion. Let R be a parabolic subgroup of G that is subordinate to the maximal parabolic P. Then R has a Levi subgroup G_{h,P} · G_{ℓ,R}. Definition (3.5.3). The holomorphic rank of R is the ℚ-rank of G_{h,P}, and is denoted ρ_{h}(R). Thus, all parabolic subgroups subordinate to a given P have the same holomorphic rank. We use this notion now to combine (3.2.9) and (3.4.1) into a single statement: Theorem (3.5.4). The differential d_{par} of K^{•} decomposes as d_{par} = d^{ℓ}+d^{h}, where d^{ℓ} preserves holomorphic rank and d^{h} lowers holomorphic rank. This induces the same for the differential d_{1} of the E_{1}-term E_{1}(K^{•}, R) of the nerve (rank) spectral sequence: d_{1} = d_{1}^{ℓ} + d_{1}^{h}. The determination of d_{1}^{ℓ} is given in (3.2.9); that of d_{1}^{h} is given in (3.4.1).

It seems right at this point to shift gears and look instead at the spectral sequence for the filtration R^{h} of K^{•} by holomorphic rank: R_{j}^{h}K^{•} is the direct sum of those terms in (3.5.1) with R of holomorphic rank ≤ j. This is clearly closed under d. We have

(3.5.5)  Gr_{t}^{R^{h}}K^{•} = ⨁_{ρ_{h}(R) = t} C^{•}(i_{R}^{∗}[W]^{can})[1 − r(R)],

on which d^{h} vanishes (compare (3.5.2)).

Though we were preoccupied with cohomology calculations in (3.1), we point out that the discussion in (3.1.1) and (3.1.4) actually give assertions in the derived category, viz.,

Proposition (3.5.6). Let P be a maximal parabolic subgroup of G, and R a parabolic subordinate to P. Let \overline{π}_{P} : Z_{Σ}(R) → M_{P, Σ(P)} be the boundary projection. Then there is a canonical isomorphism in the derived category of M_{P,Σ(P)}:

\begin{aligned}R(\overline{π}_{P})_{∗}(i_{R}^{∗}[W]^{can}) & ≈ ⨁_{w∈W^{P,p}}[V_{λ(h,w)}]^{can} ⊗ C^{•}(Γ_{ℓ,R}, V_{λ(ℓ,w)})[−ℓ(w)] \\ & ≈ ⨁_{w∈W^{R,p}}[V_{λ(h,w)}]^{can} ⊗ C^{•}(Γ_{ℓ,R}^{red}, V_{λ(ℓ,w)})[−ℓ(w)] \end{aligned}.

Note that the first factor of the tensor product above is independent of R (for R subordinate to P). If we combine (3.5.6) with (3.5.5), we obtain something “familiar”:

Corollary (3.5.7). There is an isomorphism in the derived category of M_{P, Σ(P)}:

Gr_{t}^{R^{h}}R(\overline{π}_{P})_{∗}K^{•} ≈ ⨁_{ρ_{h}(P)=t, w∈W^{P,p}}[V_{λ(h,w)}]^{can} ⊗ K_{c}^{•}(X(Γ_{ℓ,P}), V_{λ(ℓ,w)})[−ℓ(w)],

where

K_{c}^{•}(X(Γ_{ℓ,P}), V_{λ(ℓ,w)}) = ⨁_{Π(R)=P} C^{•}(Γ_{ℓ,R}, V_{λ(ℓ,w)})

is a cochain complex for the cohomology of the pair (\overline{X}(Γ_{ℓ,P}), ∂\overline{X}(Γ_{ℓ,P})).

As suggested by the notation, one can identify H^{•}(\overline{X}(Γ_{ℓ,P}), ∂\overline{X}(Γ_{ℓ,P}); V_{λ(ℓ,w)})) ≃ H_{c}^{•}(X(Γ_{ℓ,P}), V_{λ(ℓ,w)})). Numbering the standard maximal parabolics by omitted simple root, so that P_{i} ≻ P_{i+1}, we can now assert:

Corollary (3.5.8). In the spectral sequence for R^{h}, the E_{1}-term is

E_{1}^{p,q}(K^{•}, R^{h}) = H^{p+q}(Gr_{−p}^{R^{h}}K^{•})

= ⨁_{a,w;P ∼P_{ρ+p}}H^{a}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ H_{c}^{p+q−a−l(w)}(X(Γ_{ℓ,P}), Ṽ_{λ(ℓ,w)}).

NB—The non-trivial terms in the above are for p ≤ 0.

Next, we note that the differential d_{1} in (3.5.8) is induced by d^{h}. We can deduce the effect of d_{1} by observing that R is the composite, or convolution, of R^{h} and an evident analogously defined filtration R^{ℓ}, written

(3.5.9)  R = R^{h} ∗ R^{ℓ}.

This implies that

(3.5.10)  Gr_{j}^{R}K^{•} ≃ ⨁_{t+u=j} Gr_{t}^{R^{h}}Gr_{u}^{R^{ℓ}}K^{•}

(see [Z3, §1(6)]). We can see, from the calculations that produced (3.5.4), that d_{1} does “nothing” on the second factor of the tensor product in (3.5.7). In actuality, that means we are looking at the inclusions of \overline{X}(Γ_{ℓ,P}) in ∂\overline{X}(Γ_{ℓ,P′}), with ρ_{h}(P′) = ρ_{h}(P) − 1. Also, note that there is a cohomology equivalence by excision

⨆_{P}(\overline{X}(Γ_{ℓ,P}), ∂\overline{X}(Γ_{ℓ,P})) ≈ (∂\overline{X}(Γ_{ℓ,P′}), ∂\overline{X}(Γ_{ℓ,P′}) − ⨆_{P} X(Γ_{ℓ,P})).

Thus:

Proposition (3.5.11). In the spectral sequence for R^{h}, the differential

d_{1} : E_{1}^{p,q}(K^{•}, R^{h}) → E_{1}^{p+1,q}(K^{•}, R^{h})

is given by

⨁_{a,w;P ∼ P_{ρ+p}}{H^{a}(M_{P,Σ(P)}, [V_{λ(h,w)}]^{can}) ⊗ H_{c}^{p+q−a−l(w)}(X(Γ_{ℓ,P}), Ṽ_{λ(ℓ,w)})} ⟶^{d_{1}^{h}⊗δ}

⨁_{a,w;P′ ∼ P_{ρ+p+1}}{H^{a}(M_{P′,Σ(P′)}, i_{P (P′)}^{∗}[V_{λ(h,w)}]^{can})⊗H_{c}^{p+q+1−a−l(w)}(X(Γ_{ℓ,P′}), Ṽ_{λ(ℓ,w)})};

in the above, d_{1}^{h} is induced by d^{h} and acts between consecutive boundary components (P ≻ P′), and δ is the connecting homomorphism in the exact sequence of the triple

(\overline{X}(Γ_{ℓ,P′}), ∂\overline{X}(Γ_{ℓ,P′}), ∂\overline{X}(Γ_{ℓ,P′}) − ⨆_{P} X(Γ_{ℓ,P})),

where P runs over a finite set of conjugates of P_{ρ+p} (see [HZ2, App. to (3.5)]).

The E_{2}-term of the holomorphic rank spectral sequence, E_{2}(K^{•}, R^{h}), is, of course, isomorphic to the cohomology of (E_{1}(K^{•}, R^{h}), d_{1}). This can be determined as in the appendix below.

Appendix: Zipper products.

Let C^{•} be a cochain complex with differential d, and V^{•} a chain of vector spaces:

V^{0} ⟶^{T_{0}} V^{1} ⟶^{T_{1}} V^{2} ⟶^{T_{2}} ….

For the next definition, one really need assume only that C^{•} is a chain of vector spaces.

Definition (3.A.1). The zipper product Zip(C^{•}, V^{•}) of C^{•} and V^{•} is the chain of vector spaces

V^{0} ⊗ C^{0} ⟶^{T_{0}⊗d} V^{1} ⊗ C^{1} ⟶^{T_{1}⊗d} V^{2} ⊗ C^{2} ⟶^{T_{2}⊗d} ….

The following is easy to verify: Proposition (3.A.2). (i) If C^{•} is a complex, then Zip(C^{•}, V^{•}) is actually a complex.

(ii)  H^{i}(Zip(C^{•}, V^{•})) ≃ {(V^{i} ⊗ ker d_{i}) + (ker T_{i}) ⊗ C^{i}}/(im T_{i−1}) ⊗ (im d_{i−1}).

Our goal is to express the above in terms of the cohomology of C^{•}. We point out that (3.5.8) is the direct sum of zipper products. The calculation is based on the unremarkable: Lemma (3.A.3). Let A and B be subspaces of a vectorspace, and let E be a subspace of both. Then there is a short exact sequence

0 ⟶ A/E ⟶ (A + B)/E ⟶ (A + B)/A ≃ B/(A ∩ B) ⟶ 0.

In our case (3.A.2, (ii)), we compute

A/E = (V^{i} ⊗ ker d_{i})/(im T_{i−1}) ⊗ (im d_{i−1}) ≃ (im T_{i−1}) ⊗ H^{i}(C^{•}) ⊕ (Ṽ^{i} ⊗ ker d_{i}), where Ṽ^{i} is a complement to im T_{i−1} in V^{i}. On the other hand,

B/(A ∩ B) = {(ker T_{i}) ⊗ C^{i}}/{[(ker T_{i}) ⊗ C^{i}] ∩ [V^{i} ⊗ (ker d_{i})]} = (ker T_{i}) ⊗ (C^{i}/ ker d_{i}). Putting these together, we obtain: Proposition (3.A.4). H^{i}(Zip(C^{•}, V^{•})) ≃ {(im T_{i−1}) ⊗ H^{i}(C^{•})} ⊕ {(coker T_{i−1}) ⊗ (ker d_{i})} ⊕ {(ker T_{i}) ⊗ (C^{i}/ ker d_{i})}.

(3.6) Differentials and automorphic forms. It remains to express the E_{1} terms and the differentials d_{1} in terms of automorphic forms. The terms E_{1}^{1,s} have already been treated in [HZ1, §4.2]. The general case is completely analogous, and we will simply state the result. However, Franke’s theorem [Fr1] justifies replacing slowly increasing smooth functions by automorphic forms in the Lie algebra cohomology groups, and we will do so. We warn the reader that [Fr1] only treats (topological) cohomology with local coefficients. The extension to coherent cohomology is contained in an unpublished early version of [Fr1].

Suppose H is a reductive algebraic group over ℚ, P ⊂ H a parabolic subgroup, N ⊂ P its unipotent radical, L = P/N the Levi quotient. The constant term map c_{H,P} : A(H) → A(L) is defined as usual:

(3.6.1)  c_{H,P} (f)(g) = ∫_{N(ℚ)\N(A)} f(ng)dn,

where dn is the right-invariant measure on N(ℚ)\N(A) with total mass 1. If now R′ ⊂ R is a pair of parabolic subgroups of G, with Levi quotients L_{R′} and L_{R}, respectively, we write c_{R,R′} for c_{L_{R},L_{R′}}.

As in [HZ1, §4], we write I_{R}^{G} = Ind_{R(A_{f})}^{G(A_{f})}, when R is a parabolic subgroup of G. Given R subordinate to the maximal parabolic P, we write 𝔭_{h,R}^{−} = 𝔭_{h,P}^{−}. Let

(3.6.2)  𝔤_{ℓ,R} = 𝔨_{ℓ,R} ⊕ 𝔭̃_{ℓ,R}

be the (complex) Cartan decomposition. As in [HZ1, §4.2], we write

𝔭̃_{ℓ,R} = 𝔞_{R} ⊕ 𝔭_{ℓ,R},

where 𝔞_{R} is the Lie algebra of a split component A_{R} of G_{ℓ} containing the center Z_{G} of G. Finally, write 𝔰_{R} instead of 𝔰_{p,R}, and let 𝔲_{P} (R) denote the Lie algebra of the unipotent radical of R_{ℓ,P}. Then we can decompose

(3.6.3)  𝔰_{R}⟶^{∼}𝔰_{P} ⊕ 𝔲_{P} (R)

Recall from [HZ1, (3.10.3.2),(4.2.6)] that we can use the Cayley transform to identify

𝔭^{−} ⟶^{∼} 𝔭_{h,P}^{−} ⊕ 𝔲_{P,ℂ} ⊕ 𝔳_{P}^{−}.

It is better for our purposes to use the further identification

(3.6.4)  𝔭^{−} ⟶^{∼} 𝔭_{h,P}^{−} ⊕ 𝔭̃_{ℓ,P} ⊕ 𝔰_{P}

derived from [HZ1, (1.8.6)]. Now projection of Lie(R_{ℓ,P}) = 𝔤_{ℓ,R} ⊕ 𝔲_{P} (R) onto 𝔭̃_{ℓ,P} via the Cartan decomposition for 𝔤_{ℓ,P} defines an isomorphism of K_{ℓ,R}-modules

(3.6.5)  𝔭̃_{ℓ,R} ⊕ 𝔲_{P} (R)⟶^{∼}𝔭̃_{ℓ,P} .

Combining (3.6.3–3.6.5), we obtain an isomorphism of K_{h,R} · K_{ℓ,R} · A_{R}(ℝ)-modules

(3.6.6)  𝔭^{−} ⟶^{∼} 𝔭_{h,P}^{−} ⊕ 𝔭̃_{ℓ,R} ⊕ 𝔰_{R}.

Now

Theorem (3.6.7). (i) For each r, s, there is an isomorphism

E_{1}^{r,s}⟶^{∼} ⨁_{r(R)=r+1} I_{R}^{G} { ⨁_{w∈W^{R,p}}[∧^{•}(𝔭_{h,R}^{−} ⊕ 𝔭̃_{ℓ,R})^{∗} ⊗ A(L_{R}) ⊗ H^{•}(𝔰_{R}, V_{λ})]^{K_{h}·K_{ℓ,R}·A_{R}(ℝ)}]}^{s}

(on the right-hand side: terms of total degree s).

(ii) For each r, s, there is a commutative diagram

E_{1}^{r,s} ⟶^{∼}  ⨁_{r(R)=r+1} I_{R}^{G} { ⨁_{w∈W^{R,p}}[∧^{•}(𝔭_{h,R}^{−} ⊕ 𝔭̃_{ℓ,R})^{∗} ⊗ A(L_{R}) ⊗ H^{•}(𝔰_{R}, V_{λ})]^{K_{h}·K_{ℓ,R}·A_{R}(ℝ)}]}^{s}
d↓  ↓  ⨁_{R′⊊R,r(R′)=r(R)+1=r+2} c_{R,R′}
E_{1}^{r+1,s} ⟶^{∼} ⨁_{r(R′)=r+2} I_{R′}^{G} { ⨁_{w∈W^{R′,p}}[∧^{•}(𝔭_{h,R′}^{−} ⊕ 𝔭̃_{ℓ,R′})^{∗} ⊗ A(L_{R′}) ⊗ H^{•}(𝔰_{R′}, V_{λ})]^{K_{h}·K_{ℓ,R′} ·A_{R′} (ℝ)}]}^{s} .

Here the constant term maps are applied to functions, and the map on coefficients:

[∧^{•}(𝔭_{h,R}^{−} ⊕ 𝔭̃_{ℓ,R})^{∗} ⊗ H^{•}(𝔰_{R}, V_{λ})] → [∧^{•}(𝔭_{h,R′}^{−} ⊕ 𝔭̃_{ℓ,R′})^{∗} ⊗ H^{•}(𝔰_{R′}, V_{λ})]

is obtained by applying (3.6.6) to identify

∧^{•}(𝔭_{h,R}^{−} ⊕ 𝔭̃_{ℓ,R} ⊕ 𝔰_{R})^{∗}⟶^{∼} ∧^{•} (𝔭^{−})^{∗},

and likewise for R′.

Proof. That this holds for the d_{R,R′} with R and R′ subordinate to the same maximal parabolic follows from Corollary 3.2.9 and [Sch,1.10]. For R and R′ subordinate to distinct maximal parabolics, this follows from Proposition 3.4.1 and diagram (4.2.4) of [HZ1].

4. Hodge theoretic applications

In this Section, we will tie together the results and methods of [HZ1] (as generalized here in §3) with the Hodge theoretic considerations from [HZ2].

(4.1) The Hodge filtration for boundary cohomology. Let (G, X) be a pair defining a (pure) Shimura variety, E a finite-dimensional vector space over ℚ, and ρ : G → GL(E) a homomorphism of algebraic groups. This determines a compatible family of local systems Ẽ on the associated complex varieties Sh(G, X)(ℂ). Moreover, the space X, by its nature (see (1.1)), defines a family of variations of Hodge structure on Sh(G, X)(ℂ), with underlying local system Ẽ.

Let M = M_{Γ} denote a connected component of Sh(G, X)(ℂ). Let j : M → M_{Σ} be the inclusion of M in a smooth toroidal compactification, and let Z_{Σ} denote the corresponding boundary divisor ∂M_{Σ}, a divisor with normal crossings in M_{Σ}. This implies that the holomorphic logarithmic complex,

(4.1.1)  Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ Ẽ^{can}

is quasi-isomorphic to Rj_{∗}Ẽ. In the above, Ẽ^{can} denotes the canonical extension of the flat vector bundle E determined by E, as in [HZ1, 4.4]. The complex (4.1.1) is equipped with the decreasing Hodge filtration F, and the increasing weight filtration W induced from one on Rj_{∗}Ẽ. These complete the data of a cohomological mixed Hodge complex that induces, upon taking hypercohomology, the mixed Hodge structure on H^{•}(M, Ẽ).^{7}

Though the description of W on (4.1.1) is rather complicated, it is easy to specify F, and we will study the latter. Let {F^{p}Ω_{M}^{•}(log Z_{Σ})} be the usual Hodge filtration, given simply by truncation from below, and let {F^{s}} be the filtration of E that gives the variation of Hodge structure. Then the filtration F of (4.1.1) is given by the tensor product of these two:

(4.1.2)  F^{p}(Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ Ẽ^{can}) = ∑_{r+s=p} F^{r}Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ (F^{s})^{can}.

For this, there is a general definition of (F^{s})^{can} for a variation of Hodge structure, given by j_{∗}F^{s} ⋂ Ẽ^{can}. In the present case, each F^{s} is an automorphic vector bundle and (F^{s})^{can} coincides with the canonical extension in the sense of automorphic vector bundles (see [HZ1, 3.2]). Because (Ω_{M}^{p})^{can} is Ω_{M_{Σ}}^{p}(log Z_{Σ}), we can rewrite (4.1.2) as:

(4.1.3)  F^{p}(Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ Ẽ^{can}) = ∑_{r+s=p}(F^{r}Ω_{M}^{•} ⊗ F^{s})^{can}.

By Hodge theory [De2, (8.1.9)], the spectral sequence for F degenerates at E_{1}.

There are parallel assertions for deleted neighborhood cohomology. For each admissible parabolic subgroup R of G, let i_{R} : Z_{Σ}(R) → M_{Σ} denote the inclusion of the R-stratum of ∂M_{Σ}. The complex

(4.1.4)  C_{dn}^{•}(Z_{Σ}(R), Ẽ) = i_{R}^{∗}Rj_{∗}Ẽ

is called the deleted neighborhood complex for Z_{Σ}(R) in M_{Σ}, for its hypercohomology is naturally isomorphic to

(4.1.5)  H_{dn}^{•}(Z_{Σ}(R), Ẽ) = H^{•}(N(R) − Z_{Σ}(R), Ẽ),

where N(R) is a regular neighborhood of Z_{Σ}(R) in M_{Σ}. For R′ parabolic in R, let

i_{R′,R} : Z_{Σ}(R′) → Z_{Σ}(R)

7It may be useful to keep in mind that this remains true for admissible variations of mixed Hodge structure [Sa2].

denote the inclusion; note that the evident relation i_{R′} = i_{R′,R} ∘ i_{R} yields:

(4.1.6)  C_{dn}^{•}(Z_{Σ}(R′), Ẽ) ≈ i_{R′,R}^{∗}C_{dn}^{•}(Z_{Σ}(R), Ẽ).

It is not hard to see that the following complex is quasi-isomorphic to C_{dn}^{•}(Z_{Σ}(R), Ẽ):

(4.1.7)  i_{R}^{∗}(Ω_{M_{Σ}}^{•}(log Z) ⊗ Ẽ^{can}),

the complex of locally-free sheaves on Z_{Σ}(R) given by restricting (4.1.1). This complex inherits a filtration F from (4.1.2). Again, for general reasons, the filtration F induces the Hodge filtration of the natural mixed Hodge structure on H_{dn}^{•}(Z_{Σ}(R), Ẽ) [Sa3]; and the spectral sequence for F degenerates at E_{1}.

We now introduce the notation

(4.1.8)  DR(M, Ẽ) = Ω_{M}^{•} ⊗ Ẽ ≃ C^{•}(𝔭^{+}, E),

so that, for instance, (4.1.1) becomes DR(M, Ẽ)^{can} (cf. (4.1.3)). This is a complex of automorphic vector bundles that are fully decomposed only when the representation ρ is trivial. We need to determine the direct images of deleted neighborhood complexes, first in the unfiltered category (the filtered version will be deduced later, in (4.3)): Proposition (4.1.9). Let P be a maximal ℚ-parabolic subgroup of G, and R a parabolic subordinate to P. Let \overline{π}_{P} : Z_{Σ}(R) → (M_{P})_{Ξ} be the boundary projection. Then there is a canonical isomorphism in the derived category of (M_{P})_{Ξ}:

R(\overline{π}_{P})_{∗}(i_{R}^{∗})\{DR(M, Ẽ)\}^{can} ≈ \{DR(M_{P} , C̃^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)))\}^{can}.

Proof. Using the composite \overline{π}_{P} = \overline{π}_{1} ∘ \overline{π}_{2}, as per (1.4.2), we calculate that:

R(\overline{π}_{P})_{∗}(i_{R}^{∗}\{DR(M, Ẽ)\}^{can}) ≈ R\overline{π}_{1,∗}(\overline{π}_{2,∗}i_{R}^{∗}C^{•}(𝔭^{+}, E)^{can})  (by (4.1.8))
≈ C^{•}(Γ_{ℓ,R}, C^{•}(𝔰′_{P} , (C^{•}(𝔭^{+}, E))))^{can} (by (2.7.5) and (3.5.6))
≈ C^{•}(Γ_{ℓ,R}, C^{•}(𝔰′_{P} ⊕ 𝔭^{+}, E))^{can}  (by Remark 3.1.9),

where 𝔰𝔩_{2}′_{P} denotes the complex conjugate of 𝔰𝔩_{2}_{P}. The Cayley transform switches us from G to G_{h,P} as follows:

R(\overline{π}_{P})_{∗}(i_{R}^{∗})\{DR(M, Ẽ)\}^{can}) ≈ C^{•}(Γ_{ℓ,R}, C^{•}(𝔰′_{P} ⊕ 𝔭^{+}, E))^{can}  (repeated from above)
≈ C^{•}(Γ_{ℓ,R}, C^{•}(𝔭_{h,P}^{+} ⊕ 𝔴_{P} , E))^{can}  (induced by \overline{c}_{P})
≈ C^{•}(Γ_{ℓ,R}, C^{•}(𝔭_{h,P}^{+} , C^{•}(𝔴_{P} , E)))^{can}  ((3.1.9) again)
≈ C^{•}(𝔭_{h,P}^{+} , C^{•}(Γ_{ℓ,R}, C^{•}(𝔴_{P} , E)))^{can}  (as G_{h,P} and G_{ℓ,R} commute)
≈ {DR(M_{P} , C̃^{•}(Γ_{ℓ,R}, C^{•}(𝔴_{P} , E)))}^{can}  (equivalent formula)
≈ {DR(M_{P} , C̃^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)))}^{can}(follows by Kostant; cf. (2.7.5)).

(4.1.10) Remarks. i) Recall (cf. [HZ1, 1.8.3]) that conjugation by c_{P}^{−1} takes a subgroup K_{t} of K_{p} onto G_{ℓ}, and transforms the adjoint action of K_{t} on 𝔰_{P} ⊕ 𝔭^{−} to the action of G_{ℓ} on 𝔭_{h,P}^{−} ⊕ 𝔴_{P}. Thus the action of Γ_{ℓ,R} on C^{•}(𝔰_{P} ⊕ 𝔭^{+}, E) in the first line above comes from the identification of Γ_{ℓ,R}, via \overline{c}_{P} = c_{P}^{−1}, with a subgroup of K_{p}, or equivalently, the identification

C^{•}(𝔰′_{P} ⊕ 𝔭^{+}, E) ≃ C^{•}(𝔭_{h,P}^{+} ⊕ 𝔴_{P} , E).

The above is the basis of the calculations in [HZ1] that underlie this isomorphism (see [HZ1, Lemma 3.5.11]). The action on the second line of the above series of quasi-isomorphisms, by contrast, is via the usual (adjoint) action of G_{ℓ,R} on 𝔴_{P}; G_{ℓ,R} and \overline{c}_{P} both act as the identity on 𝔭_{h,P}^{+} .

Moreover, for P′ ≺ P, one has c_{P′} = c_{P} ∘ c_{P′,P}, where c_{P′,P} denotes the Cayley transform for X̂_{P′} as a boundary component of X̂_{P}.

ii) Note that we have elected not to make use of the Kostant decomposition of H^{•}(𝔰_{P} , V) (thus also its complex conjugate), utilized in [HZI, 3.6] and Section 3 here. The conclusion of the above result can be viewed as the extension of (3.5.6) to special complexes that are not fully decomposed. Also, the final answer is, as one expects after [HZ1], independent of the choice of toroidal compactification. Introducing the Kostant decomposition,

(4.1.10.1)  H^{•}(𝔴_{P} , E) = ⨁_{w∈W^{P}} E_{μ(h,w)} ⊗ E_{μ(ℓ,w)}[−l(w)],

we can rewrite the formula of (4.1.9) as:

R(\overline{π}_{P})_{∗}i_{R}^{∗}Rj_{∗}Ẽ ≈ ⨁_{w∈W^{P}} R(j_{P})_{∗}Ẽ_{μ(h,w)} ⊗ H^{•}(Γ_{ℓ,R}, E_{μ(ℓ,w)})[−l(w)],

≈ {DR(M_{P} , L̃_{R})}^{can} (4.1.10.2)

where j_{P} : M_{P} ↪ (M_{P})_{Ξ} is the inclusion, and L_{R} = H^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E))).

iii) Throughout, we can express the Γ_{ℓ,R}-cohomology as Γ_{ℓ,R}^{red}-cohomology, using the maneuver that produced (3.1.8). We would then write

L_{R} = H^{•}(Γ_{ℓ,R}^{red}, H^{•}(𝔴_{R}, E)).

Next, let R′ = R ∩ P′, with P′ maximal. We consider the restriction mappings from Z_{R} to Z_{R′} in conjunction with Proposition (4.1.9). If P′ ≻ P, then R′ is also subordinate to P. In that case,

(4.1.11.1)  R(\overline{π}_{P})_{∗}((i_{R}^{∗})\{DR(M, Ẽ)\}^{can}) ⟶ R(\overline{π}_{P})_{∗}(i_{R′}^{∗}\{DR(M, Ẽ)\}^{can})

is given tautologically:

(4.1.11.2) {DR(M_{P} , C̃^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)))}^{can} ⟶ {DR(M_{P} , C̃^{•}(Γ_{ℓ,R′}, H^{•}(𝔴_{P} , E)))}^{can},

which is induced by the morphism of local systems coming by restriction on the Γ_{ℓ}’s. After grading for F, the above is essentially equivalent to what we discussed in (3.2). On the other hand, if P′ ≺ P, then R′ is subordinate to P′. In that case, one reverts to the determination in (3.4).

The treatment of d^{h} has the same problem and the same remedy as before. Consider the diagram:

(4.1.12)  \begin{aligned}Z_{Σ}(R) & ⟵^{i_{R′,R}} & Z_{Σ}(R′) \\ \overline{π}_{P} ↓ & & ↓^{τ} \\ M_{P,Σ(P)} & ⟵^{i_{P (P′)}} & Z_{Σ(P)}(P (P′)) \\ ↓ & \overline{π}_{P,P′} \\ & & M_{P′,Σ(P′)} \end{aligned}

Let L_{R}^{•} = C^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)). Then the calculation:

R(\overline{π}_{P′})_{∗}(i_{R′}^{∗}\{DR(M, Ẽ)\}^{can}) = R(\overline{π}_{P,P′})_{∗}Rτ_{∗}(i_{R′,R}^{∗} i_{R}^{∗})\{DR(M, Ẽ)\}^{can}
≈ R(\overline{π}_{P,P′})_{∗}i_{P (P′)}^{∗}R(\overline{π}_{P})_{∗}i_{R}^{∗}\{DR(M, Ẽ)\}^{can} (from (2.7.11))
≈ R(\overline{π}_{P,P′})_{∗}i_{P (P′)}^{∗}\{DR(M_{P} , L̃_{R})\}^{can} (by (4.1.9))

≈ {DR(M_{P′}, C̃^{•}(Γ_{ℓ,P (P′)}, H^{•}(𝔲_{P (P′)}, L_{R})))}^{can} (4.1.13)

= {DR(M_{P′}, C̃^{•}(Γ_{ℓ,P (P′)}, H^{•}(𝔲_{P (P′)}, H^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)))}^{can}
≃ {DR(M_{P′}, C̃^{•}(Γ_{ℓ,R′}, H^{•}(𝔴_{P′}, E)))}^{can} (as in (3.1.6))

recovers the direct computation of R(\overline{π}_{P′})_{∗}i_{R′}^{∗}\{DR(M, Ẽ)\}^{can}, because of (4.1.10) and the way in which the Cayley transforms entered. We now see that d^{h} is induced by restriction on the M_{P} ’s.

When one grades (4.1.2) for F, the differentials become linearized, and each Gr_{F}^{p} (Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ Ẽ^{can}) becomes a complex of canonically-extended, fully decomposed automorphic vector bundles, placing us in the context of (3.2). For (4.1.7),

(4.1.14)  i_{R}^{∗}(Gr_{F}^{p} (Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ Ẽ^{can}))

becomes the restriction of the same to Z_{Σ}(R).

We briefly recall the topological nerve spectral sequence that is of fundamental interest, which we treated in [HZ2, (3.5.4)]; it is the one associated to the local system Ẽ and the covering of the Borel-Serre boundary by its closed faces, \{\overline{e′(P)} : P maximal}:

(4.1.15)  E_{1}^{r,s} = ⨁_{r(R)=r+1} H^{s}(\overline{e′(R)}, Ẽ) ⇒ H^{r+s}(∂\overline{M}, Ẽ).

By [HZ2, (3.5.5)], the above coincides with the corresponding spectral sequence for deleted neighborhood cohomology on M_{Σ}, viz.,

(4.1.16)  E_{1}^{r,s} = ⨁_{r(R)=r+1} H_{dn}^{s}(Z_{Σ}(R), Ẽ) ⇒ H_{dn}^{r+s}(Z_{Σ}, Ẽ).

We can use (4.1.7) and (4.1.8) to rewrite (4.1.16) in the form (4.1.17)

E_{1}^{r,s} = ⨁_{r(R)=r+1} ℍ^{s}(Z_{Σ}(R), i_{R}^{∗}DR(M, Ẽ)^{can}) ⇒ ℍ^{r+s}(Z_{Σ}, i^{∗}DR(M, Ẽ)^{can}),

where ℍ indicates hypercohomology. This is a spectral sequence of mixed Hodge structures (see [HZ2, (5.5.2)]; cf. (4.5.2) here). A morphism of mixed Hodge structures is completely determined by its gradation for the Hodge filtration F, so we can grade for F without losing information about kernels and images; this reduces such questions about the spectral sequence (4.1.12) to analogous considerations for spectral sequences of the form (3.2.1). Thus,

(4.1.18) Gr_{F}^{•} E_{1}^{r,s} = ⨁_{r(R)=r+1} Gr_{F}^{•} ℍ^{s}(Z_{Σ}(R), i_{R}^{∗}DR(M, Ẽ)^{can})

⇒ Gr_{F}^{•} ℍ^{r+s}(Z_{Σ}, i^{∗}DR(M, Ẽ)^{can}).

By basic Hodge theory, we can write (4.1.18) as

Gr_{F}^{•} E_{1}^{r,s} = ⨁_{r(R)=r+1} ℍ^{s}(Z_{Σ}(R), Gr_{F}^{•} {i_{R}^{∗}DR(M, Ẽ)^{can}})

⇒ ℍ^{r+s}(Z_{Σ}, Gr_{F}^{•} {i^{∗}DR(M, Ẽ)^{can}}).

Using (4.1.14) and (4.1.16), we obtain:

(4.1.19) Gr_{F}^{•} E_{1}^{r,s} = ⨁_{r(R)=r+1} H^{s}(Z_{Σ}(R), i_{R}^{∗}{Gr_{F}^{•} H^{•}(𝔭^{+}, E)}^{can})

⇒ H^{r+s}(Z_{Σ}, i^{∗}{Gr_{F}^{•} H^{•}(𝔭^{+}, E)}^{can}).

(4.2) The dual Bernstein-Gelfand-Gelfand complex. Following Faltings [F], we can replace the de Rham complex DR(M, Ẽ) by a quasi-isomorphic filtered subcomplex, the dual BGG complex BGG(M, Ẽ), which is minimal in the sense that each degree is given by a sum of irreducible automorphic vector bundles. After taking cohomology, the quasi-isomorphism between BGG(M, Ẽ) and DR(M, Ẽ) recovers the results of [Z1]. Because its terms are irreducible automorphic vector bundles, the dual BGG complex has particularly nice rationality and integrality properties. Moreover, its cohomology is directly expressible in terms of automorphic forms

We use the formalism introduced in [H5, §2], but with slightly different notation. We assume first that E is an absolutely irreducible representation of G, with highest weight Λ ∈ 𝔥^{∗} (see (0.3)); the irreducibility hypothesis will be removed at the end of the section. Let W^{1} denote the set of Kostant representatives for the parabolic subalgebra 𝔓^{+} (see (0.3)). For any t ∈ W^{1}, we let Λ(t) = t(Λ + ρ) − ρ, where ρ is the half-sum of positive roots. Then Λ(t) is the highest weight of an irreducible representation of K_{p}, hence defines an irreducible automorphic vector bundle which we denote BGG^{t}(M, Ẽ). For 0 ≤ i ≤ dim_{ℂ}(M), let

BGG^{i}(M, Ẽ) = ⨁_{ℓ(t)=i} BGG^{t}(M, Ẽ).

The BGG^{i}(M, Ẽ) form a complex BGG(M, Ẽ) in which the differentials are algebraic differential operators, and there is a canonical inclusion [F]

(4.2.1)  BGG(M, Ẽ) ↪ DR(M, Ẽ).

In (4.2.1), one places the Hodge filtration F on the left-hand side in the usual way for homogeneous vector bundles (see [H5,(2.2.3)]), and we have likewise the Hodge filtration from (4.1) on the right-hand side.

Theorem (4.2.2) [F]. The inclusion (4.2.1) is a filtered quasi-isomorphism that extends to a filtered quasi-isomorphism of complexes of canonical extensions.

Remark. It is implicit in the second part of the above theorem that the complex BGG(M, Ẽ)^{can} is functorial with respect to change of toroidal compactification. This follows easily from Proposition 1.3.7, or alternatively from the functoriality of the canonical extension of the de Rham complex.

Theorem (4.2.2) yields immediately:

Corollary (4.2.3). There is a spectral sequence

E_{1}^{p,q} = H^{p+q}(Gr_{F}^{p} BGG(M, Ẽ)^{can}) ⇒ H^{p+q}(M, Ẽ)

that degenerates at E_{1}. The filtration induced by F on the abutment is the Hodge filtration in its mixed Hodge structure.

Using (4.2.2) on both sides of (4.1.9), we obtain the following reformulation of the latter:

Corollary (4.2.4). In the situation of (4.1.9), there is a canonical isomorphism in the derived category of (M_{P})_{Ξ}:

R(\overline{π}_{P})_{∗}(i_{R}^{∗})\{BGG(M, Ẽ)\}^{can} ≈ \{BGG(M_{P} , C̃^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)))\}^{can}.

We present below a more concrete way to see this quasi-isomorphism.

We begin with some general considerations involving Weyl group combinatorics. Let G be a reductive group as in (0.2), with chosen set Φ^{+} of positive roots, and let P, Q ⊂ G denote two standard parabolics; let R = P ∩ Q. Let 𝔴_{P}, 𝔴_{Q}, 𝔴_{R} denote the Lie algebras of the unipotent radicals of P, Q, and R, respectively; Similarly, let L_{?}, for ? = P, Q, R, be a compatible set of Levi factors. Then we can decompose

(4.2.5)  𝔴_{R} = 𝔴_{P Q} ⊕ 𝔴_{P}^{Q} ⊕ 𝔴_{Q}^{P}

in such a way that

𝔴_{P} = 𝔴_{P Q} ⊕ 𝔴_{P}^{Q}, 𝔴_{Q} = 𝔴_{P Q} ⊕ 𝔴_{Q}^{P}

(so 𝔴_{P Q} = 𝔴_{P} ∩ 𝔴_{Q}) and Φ_{R} = Φ_{P Q} ⊔ Φ_{P}^{Q} ⊔ Φ_{Q}^{P}; here Φ_{R}, Φ_{P Q}, Φ_{P}^{Q}, Φ_{Q}^{P} denote the sets of positive roots in 𝔴_{R}, etc. Let W^{?} ⊂ W, for ? = P, Q, R, denote the corresponding sets of Kostant representatives, and let W_{?} ⊂ W be the Weyl group of L_{?}. Finally, L_{P} ∩ R = L_{P} ∩ Q is parabolic in L_{P} and we let W_{P}^{R} = W_{P}^{Q} denote the Kostant representatives in W_{L_{P}} = W_{P} relative to W_{L_{P} ∩R}; define W_{Q}^{R} = W_{Q}^{P} similarly. As in Lemma 3.1.6, we have canonical decompositions

(4.2.6)  W^{R} = W_{P}^{R} · W^{P} = W_{Q}^{R} · W^{Q}.

We denote the elements of W^{R} by ω, and those of W_{P}^{Q}, W^{P}, W_{Q}^{P}, W^{Q} by a, b, s, and t, respectively. For any ω ∈ W^{R} we write

Φ_{P Q}(ω) = ω(Φ^{+}) ∩ −Φ_{P Q}; Φ_{P}^{Q}(ω) = ω(Φ^{+}) ∩ −Φ_{P}^{Q}; Φ_{Q}^{P}(ω) = ω(Φ^{+}) ∩ −Φ_{Q}^{P}.

Then

(4.2.7)  ℓ(ω) = |Φ_{P Q}(ω)| + |Φ_{P}^{Q}(ω)| + |Φ_{Q}^{P}(ω)|.

If ω = ab = st by the factorizations of (4.2.6), we then have

(4.2.8)  ℓ(ω) = ℓ(a) + ℓ(b) = ℓ(s) + ℓ(t),

(4.2.9) ℓ(a) = |Φ_{P}^{Q}(ω)|, ℓ(b) = |Φ_{P Q}(ω)| + |Φ_{Q}^{P}(ω)|; ℓ(s) = |Φ_{Q}^{P}(ω)|, ℓ(t) = |Φ_{P Q}(ω)| + |Φ_{P}^{Q}(ω)|.

Finally, if ω, ω′ ∈ W, we write ω→ω′ if there exists γ ∈ Φ^{+} such that ω = r_{γ}ω′ and ℓ(ω) = ℓ(ω′) + 1; here r_{γ} is the reflection attached to γ.

Lemma (4.2.10). . Let ω, ω′ ∈ W^{R}, with factorizations ω = ab = st, ω′ = a′b′ = s′t′ as above. Suppose ω→ω′. Suppose further that b = b′. Then s = s′ and t→t′.

Proof. It follows from (4.2.8) and (4.2.9) that ℓ(t) = ℓ(t′) + 1. Indeed, Φ_{P Q}(ω) = Φ_{P Q}(b) = Φ_{P Q}(ω′). Suppose we know that s = s′. Writing st = r_{γ}st′, we see that t = r_{s^{−1}γ}t′, hence t→t′. So it remains to prove s = s′. For this it suffices to show that Φ_{P}^{Q}(s) = Φ_{P}^{Q}(s′), since both s and s′ preserve 𝔴_{Q}. But

Φ_{P}^{Q}(s) = Φ_{P}^{Q}(ω) = Φ_{P}^{Q}(b), Φ_{P}^{Q}(s′) = Φ_{P}^{Q}(ω′) = Φ_{P}^{Q}(b′),

where the first equalities follow from the definition of W^{Q} and the second from the fact that W_{P}^{Q} preserves 𝔴_{P}. The assertion is now clear.

Let 𝔤, 𝔭, 𝔮, 𝔯 denote the Lie algebras of G, P, Q, and R respectively. Letting E and Λ be as above, we consider the generalized Bernstein-Gelfand-Gelfand resolutions of E with respect to the parabolic subalgebras we have introduced. Thus for ω ∈ W^{R} let

(4.2.11)  𝕄_{R}(Λ(ω)) = 𝔘(𝔤) ⊗_{𝔘(𝔯)} E(Λ(ω)),

with Λ(ω) defined as above and E(Λ(ω)) the finite-dimensional L_{R} module with highest weight Λ(ω), extended as usual to an irreducible 𝔘(𝔯) module. The generalized BGG complex BGG^{R}(E) of E, relative to 𝔯, is the complex (4.2.12)

… → ⨁_{ω∈W^{R},ℓ(w)=i+1}𝕄_{R}(Λ(ω)) →^{d_{i}^{R}} ⨁_{ω∈W^{R},ℓ(ω)=i}𝕄_{R}(Λ(w)) → … →𝕄_{R}(Λ)→0.

Here d_{i}^{R} is the differential constructed in [Le, §4]. It can be written as a sum

(4.2.13)  d_{i}^{R} = ∑_{ω,ω′∈W^{R},ω→ω′,ℓ(ω′)=i} d_{ω,ω′}^{R},

where d_{ω,ω′}^{R} : 𝕄_{R}(Λ(ω))→𝕄_{R}(Λ(ω′)). These differentials can be defined uniquely up to sign ([RC, Lemma 10.5]), and the natural surjection 𝕄_{R}(Λ)→E (defined up to scalar multiples) defines a quasi-isomorphism BGG^{R}(E) ≈ E in the category of 𝔘(𝔤)-modules. It follows from the results of [RC, §§10-11] that changing the sign of one differential changes all signs simultaneously, and that the different choices of sign yield isomorphic complexes.

Let \overline{P} = L_{P} \overline{N}_{P} be the parabolic opposite to P, with Lie algebra \overline{𝔭} = 𝔩 ⊕ \overline{𝔴}_{P}. As 𝔘(\overline{𝔭})-module we have 𝕄_{R}(Λ(ω)) = 𝔘(\overline{𝔭}) ⊗_{𝔘(𝔩_{P} ∩𝔯)} E(Λ(ω)). For any irreducible finite-dimensional L_{P} -module V, we let BGG^{L_{P} ∩R}(V) denote the generalized BGG resolution of V relative to L_{P} ∩ R, and define the maps d_{a,a′}^{L_{P} ∩R} by analogy with (4.2.13).

Now, BGG^{R}(E) is \overline{𝔫}_{P} -acyclic, hence the complex H_{0}(\overline{𝔴_{P}} , BGG^{R}(E)) computes H_{•}(𝔴_{P} , E), via the quasi-isomorphism BGG^{R}(E) ≈ E. For any ω we have natural isomorphisms

(4.2.14)  H_{0}(\overline{𝔴}_{P} , 𝕄_{R}(Λ(ω)))⟶^{∼}𝔘(𝔩_{P}) ⊗_{𝔘(𝔩_{P} ∩𝔯)} E(Λ(ω)).

We let 𝕄(Λ(ω); L_{P}) denote the right-hand side of (4.2.14). Then

Proposition (4.2.15). (i) There is a canonical isomorphism of complexes

H_{0}(\overline{𝔫}_{P} , BGG^{R}(E))⟶^{∼} ⨁_{b∈W^{P}} BGG^{R(P)}(E(Λ(b))[ℓ(b)].

Here E(Λ(b)) is the irreducible representation of L_{P} with highest weight Λ(b).

(ii) More precisely, for any pair a, a′ ∈ W_{P}^{Q} = W^{R(P)}, with a→a′, and for any b ∈ W^{P}, there is a pair t, t′ ∈ W^{Q} and s ∈ W_{Q}^{P} such that the differential d_{a,a′}^{R(P)} in BGG^{R(P)}(E(Λ(b))[ℓ(b)] is obtained by applying H_{0}(\overline{𝔫}_{P} , •) to d_{ω,ω′}^{R}, with ω = ab = st, ω′ = a′b = st′.

(iii) For any pair ω, ω′ ∈ W^{R} not obtained as in (ii), the morphism H_{0}(\overline{𝔫}_{P} , d_{ω,ω′}^{R}) vanishes.

Proof. We first verify (ii) and (iii). Assertion (ii) is a consequence of Lemma 4.2.10. Suppose ω→ω′ in W^{R}, and write ω = ab, ω′ = a′b′. Now it follows from the theory of Verma modules for L_{P} that the infinitesimal character separates 𝕄(Λ(ω); L_{P}) from 𝕄(Λ(ω′); L_{P}) if b ≠ b′. Thus H_{0}(\overline{𝔫_{P}} , d_{ω,ω′}^{R}) = 0 unless b = b′, and (iii) follows from (ii).

The assertion (i) follows easily from (ii) and (iii) by checking degrees. Indeed, by writing ω = ab as above, we see that the assertion is true in each degree separately, and it remains only to check that the differentials correspond (up to sign). But this follows from the explicit construction of the generalized BGG resolutions in [Le,RC].

We fix a toroidal compactification Sh(G, X)_{Σ} (omitting the left subscript for the level subgroup), with Σ assumed to be sufficiently fine for the following arguments. Fix a maximal parabolic subgroup P ⊂ G. Let P′ ⊂ P denote G_{h,P} · W_{P}, as in [HZ1, 1.2.3]. Thus there is a mixed Shimura datum (P′, X) and a partially compactified mixed Shimura variety Sh(P′, X)_{Σ′} corresponding to the P -stratum of the boundary of Sh(G, X)_{Σ}. We refer to (1.4) for the morphisms π_{1,Σ′}, π_{2,Σ′}, and let π_{Σ′} = π_{1,Σ′} ∘ π_{2,Σ′} denote the composite morphism from Sh(P′, X)_{Σ′} to M_{P,Σ(P)}, where M_{P,Σ(P)} is a toroidal compactification of the pure Shimura variety associated to (G_{h,P} , W \X). We let i_{P}^{o} denote the inclusion of the open P -stratum in the boundary, and let π : Sh(P′, X)→M_{P} be the corresponding map of uncompactified (mixed) Shimura varieties. The differentials in the complex i_{P}^{o,∗}BGG(M, Ẽ) (with M a connected component of Sh(G, X)) are differential operators of positive degree, hence not O-linear. But they are morphisms of abelian sheaves, hence give rise to functorial morphisms under R^{i}π_{∗}, for each i, and these are also differential operators. Our goal is to calculate these directly at the level of BGG(M_{P} , C̃^{•}(Γ_{ℓ,R}, H^{•}(𝔴_{P} , E))), using Proposition 4.2.4.

We apologize that the standard maximal parabolic P here will play the role of \overline{P} in the applications of Proposition 4.2.15 in what follows; this is because we are ultimately interested in homology and cohomology of 𝔴_{P}, not \overline{𝔴}_{P}. Thus, our notation is correspondingly contorted. First, let Q_{1} be the complex parabolic with Lie algebra 𝔨 ⊕ 𝔓^{−}, and \overline{Q} = Ad(c_{P}^{−1})(Q_{1}). Then, let \overline{Q} ⊂ G denote the standard complex parabolic conjugate to Q_{1} = 𝔨 ⊕ 𝔓^{−}; and Q the parabolic opposite to \overline{Q}, viz., Ad(c_{P})(𝔨 ⊕ 𝔓^{−}). Let R = \overline{P} ∩ Q, and R_{1} = Ad(c_{P})R ⊂ Q_{1}.

Via Ad(c_{P}) we may identify W^{Q} = W^{Q_{1}}, W^{R} = W^{R_{1}}. We use the same letters a, b, s, t as above to denote corresponding Kostant representatives. In particular, we denote the components of the BGG complex by BGG^{t}(Ẽ), dropping the “M” in (4.2.1). Similarly, for each q and t, we have

(4.2.16)  R^{q}π_{∗}BGG^{t}(Ẽ)⟶^{∼} ⊕_{s∈W \overline{P} ; Q ,ℓ(s)=q} [BGG^{t}(Ẽ)]^{s}.

Here if BGG^{t}(Ẽ) is the automorphic vector bundle associated to the representation E(t) of K_{p}, [BGG^{t}(Ẽ)]^{s} is the automorphic vector bundle on M_{P} associated to C^{•}(Γ_{ℓ,P} , H^{s}(𝔰′_{P} , E(t))), where H^{s} denotes the Kostant constituent of the cohomology associated to s ∈ W \overline{P} ; Q. We are abusing notation (in a harmless way) by writing in the discrete group Γ_{ℓ,P} in this adelic situation. If now ω = st = ab as in (4.2.6), then we can identify

(4.2.17)  [BGG^{t}(Ẽ)]^{s} = C^{•}(Γ_{ℓ,P} , BGG^{a}(H̃^{b}(𝔴_{P} , E))).

It then follows from (4.2.6) that

Lemma (4.2.18). The identifications (4.2.16) and (4.2.17) define an isomorphism between the constituents of the two quasi-isomorphic complexes in (4.2.4).

It remains to show that these isomorphisms are compatible with the differentials on the two sides of (4.2.4). Recall that, if E and F are vector bundles on a smooth complex variety Z, a ℂ-linear map Δ : E→F is a differential operator of order n if it can be factored as

E ⟶^{j^{n}} jet^{n}(E) → F,

where jet^{n}(E) is the bundle of n-jets of sections of E, j^{n} is the canonical differential operator, and the right-hand arrow is O_{Z}-linear. (Here the notation for jets is as in [H1].) Classification of differential operators Δ : E→F thus comes down to classification of O_{Z}-linear maps jet^{∞}(E)→jet^{∞}(F), where jet^{∞} denotes the inverse limit over the jet bundles of finite order.

For our purposes, it is more useful to look at the dual picture. Let D_{Z} denote the sheaf of differential operators on Z; we write D instead of D_{Z} when the base Z is understood. For G a vector bundle on Z, let D(G) = D_{Z}(G) denote the locally free D-module D ⊗_{O_{Z}} G^{∨}. Then to Δ as above we associate a homomorphism of D-modules

(4.2.19)  Δ^{∨} : D(F^{∨}) → D(E^{∨}).

More precisely, jet^{∞}(E) is naturally isomorphic to the O_{Z}-linear dual of D_{Z}(E^{∨}) and the dual of Δ^{∨} is the map of infinite jet bundles mentioned above. We call Δ^{∨} the linearization of Δ.

We fix a point p ∈ X compatible with our system of standard rational parabolic subgroups. If [W] is the automorphic vector bundle corresponding to the irreducible representation (ρ, W_{ρ}) of the stabilizer K_{p} of p, then D([W]) is the automorphic vector bundle corresponding to the natural representation of 𝔓_{p}^{−} on the generalized Verma module

𝔻(W_{ρ}^{∗}) = 𝔘(𝔤) ⊗_{𝔘(𝔓_{p}^{−})} W_{ρ}^{∗}.

We point out that D([W]^{can}) ≠ D([W])^{can}. However, we will only be concerned with the latter, to which logarithmic differential operators naturally extend.

We now fix a maximal standard rational parabolic P and the subgroup P′ ⊂ P to which the corresponding boundary mixed Shimura datum (P′, X) is attached. The constructions in (1.2) associate automorphic vector bundles on the associated mixed Shimura varieties to representations of the stabilizer in P′ of the fixed point p ∈ X. This stabilizer will be denoted F^{0}P′, since its Lie algebra is F^{0}Lie(P′) for the Hodge filtration associated to p; in particular, we drop the base point p from the notation. In what follows, we will identify automorphic vector bundles with the locally homogeneous vector bundles attached to representations of

F^{0}Lie(P′) = 𝔨_{h,P} ⊕ 𝔭_{h,P}^{−} ⊕ 𝔳^{−},

and we will also drop the subscript P from the above.

Without loss of generality we may assume that, for fixed x in the P -stratum, there is an étale analytic morphism from Sh(P′, X)_{Σ′} to a neighborhood of x. The pullback of D([W]) to Sh(P′, X) via this morphism is the D(•) of the pullback of [W]. This can be determined just as above, and we find that it is the automorphic vector bundle corresponding to the natural representation of F^{0}Lie(P′) on the tensor product module (4.2.20)

𝔻_{P} (W_{ρ}^{∗}) = 𝔘(Lie(P′)) ⊗_{𝔘(F^{0}Lie(P′))} W_{ρ}^{∗}⟶^{∼}𝔘(Lie(P)) ⊗_{𝔘(F^{0}Lie(P′)⊕𝔤_{ℓ})} W_{ρ}^{∗},

where the second isomorphism just serves to indicate that arithmetic subgroups of G_{ℓ} = G_{ℓ,P} act on the coefficients and commute with all homogeneous differential operators.

Now we concentrate on the map, again denoted π : Z = Sh(P′, X)_{Σ′} → M_{P}, where the subscript Σ′ corresponds to a compactification in the vertical direction. In particular the map π is smooth and proper. Henceforward, we let

E = BGG^{t′}(Ẽ), F = BGG^{t}(Ẽ),

with t→t′ in W^{Q} as above. Let Λ be the highest weight of the contragredient of E. Then the fibers at p of F^{∨} and E^{∨} are 𝔻_{P} (H^{t}(𝔭^{+}, E)^{∨}) and 𝔻_{P} (H^{t′}(𝔭^{+}, E)^{∨}), respectively. In turn, these can be identified respectively with 𝕄_{Q_{1}}(Λ(t)) and 𝕄_{Q_{1}}(Λ(t′)). The differential d_{t,t′}^{Q_{1}} : 𝕄_{Q_{1}}(Λ(t))→𝕄_{Q_{1}}(Λ(t′)) in the generalized BGG complex gives rise to the differential operator Δ_{t,t′} : E→F, corresponding to a component of the differential in BGG(Ẽ).

To determine the image of Δ_{t,t′} under R^{i}π_{∗}, we pass to linearizations. The map Δ_{t,t′}^{∨} of (4.2.19) is a morphism of D_{Z}-modules, hence it is natural to consider its image under the direct image in the category of D_{Z}-modules. Following [B4], we denote this derived functor Rπ_{+}.

Proposition (4.2.21). (i) Let G = E or F, and d = dim Z − dim M. For each q ≥ 0, there is a natural isomorphism

R^{•}π_{+}D_{Z}(G^{∨})⟶^{∼}D_{M_{P}} (R^{d−•}π_{∗}(G^{∨})).

(ii) Under this isomorphism, the image of Δ_{t,t′}^{∨} is the linearization of the image of Δ_{t,t′} via R^{d−•}π_{∗}.

Proof. The first assertion is a simple consequence of the definitions. Recall that π is smooth and proper. By definition, for any D_{Z}-module N,

Rπ_{+}N = Rπ_{∗}(D_{{M_{P} ←Z}} ⊗^{𝕃}_{D_{Z}} N).

Here D_{{}M_{P} ← Z} = π^{∗}(D_{M_{P}}) ⊗ ω_{Z/M_{P}}, where π^{∗} is inverse image in the category of O-modules and ω_{Z/M_{P}} = Ω_{Z/M_{P}}^{d} is the relative dualizing sheaf. In our case, N = D_{Z} ⊗_{O_{Z}} G^{∨} is a free D_{Z}-module, hence we can replace ⊗^{𝕃} by ⊗ above, and we have

(4.2.22)  \begin{aligned}R^{•}π_{+}D_{Z}(G^{∨}) & = R^{•}π_{∗}[π^{∗}(D_{M_{P}}) ⊗ ω_{Z/M_{P}} ⊗ G^{∨}] \\ & ⟶^{∼} D_{M_{P}} ⊗ R^{•}π_{∗}(ω_{Z/M_{P}} ⊗ G^{∨}) \\ & ⟶^{∼} D_{M_{P}} ⊗ (R^{•}π_{∗}G)^{∨} = D_{M_{P}} (R^{d−•}π_{∗}(G^{∨})) \end{aligned}

where the first isomorphism is the projection formula and the second is Serre duality. The second assertion is well-known (see [Sa1, Lemme 2.3.6]).

We rewrite the formulas (4.2.22) in terms of representation theory. Recall that

(4.2.23)  𝔴_{P,ℂ} = 𝔲_{ℂ} ⊕ 𝔳^{+} ⊕ 𝔳^{−}

as modules over 𝔨_{h}; here 𝔲 = Lie(U) with U the center of W_{P}, as usual. We write 𝔴^{+} = 𝔲_{ℂ} ⊕ 𝔳_{p}^{+}. Then the bundle ω_{Z/M_{P}} is the automorphic vector bundle on M_{P} associated to the adjoint representation of 𝔨_{h} on the highest exterior power of the holomorphic cotangent space at the image of p in M_{P}, i.e. on ∧^{d}(𝔴^{+})^{∗}. Next, π^{∗}(D_{M_{P}}) is the bundle associated to the adjoint representation of F_{0}Lie(P′) (or F_{0}Lie(P), if we want to keep track of the G_{ℓ}-action) on H_{0}(𝔴^{+}, 𝔘(Lie(P′))) (or H_{0}(𝔴^{+}, 𝔘(Lie(P))), but the addition of 𝔤_{ℓ} at this point changes nothing in the final calculation). Finally, the functor Rπ_{∗} corresponds to the standard bar complex C^{•}(𝔳^{−}, •) for Lie algebra cohomology. Thus the first line of (4.2.22) expresses Rπ_{+}D_{Z}(G^{∨}) as the automorphic vector bundle attached to

C^{•}(𝔳^{−}, H_{0}(𝔴^{+}, 𝔘(Lie(P)) ⊗_{𝔘(F^{0}Lie(P))} ∧^{d}(𝔴^{+})^{∗} ⊗ W)),

where G^{∨} is the automorphic vector bundle attached to the representation of 𝔨 on W. Since 𝔳^{−} ⊂ F^{0}Lie(P′) the action passes across the tensor product to yield

H_{0}(𝔴^{+}, 𝔘(Lie(P)) ⊗_{𝔘(F^{0}Lie(P))} C^{•}(𝔳^{−}, ∧^{d}(𝔴^{+})^{∗} ⊗ W))

Koszul duality [Kn, Theorem 6.10 (6.30)] provides a canonical identification

C^{•}(𝔳^{−},?)⟶^{∼}C_{d_{−}−•}(𝔳^{−}, ∧^{dim 𝔳^{−}}(𝔳^{−})^{∗}⊗?),

where C_{•}(𝔳^{−},?) is the standard complex for Lie algebra homology and d_{−} = dim 𝔳^{−}. Thus we can replace the last complex by

(4.2.24) H_{0}(𝔴^{+}, 𝔘(Lie(P′ · W_{2}P′_{ℂ}) ⊗_{𝔘(F^{0}Lie(P′))} C_{d_{−}−•}(𝔳^{−}, ∧^{d_{P}} (𝔴_{P})^{∗} ⊗ W)) = H_{0}(𝔴^{+}, 𝔘(Lie(P′ · W_{2}P′_{ℂ}) ⊗_{𝔘(F^{0}Lie(P′))} ∧^{d_{P}} (𝔴_{P})^{∗} ⊗ C_{d_{−}−•}(𝔳^{−}, W))

Here d_{P} = dim 𝔴_{P} and we have used (4.2.23) and the fact that 𝔳^{−}, being unipotent, has trivial determinant on 𝔴_{P}.

Remember that 𝔳^{−} acts on W via an identification with 𝔰 ⊂ 𝔨, the unipotent radical of a parabolic subalgebra we will denote \overline{𝔯}(Q). Similarly, we identify 𝔤_{ℓ} with Ad(c_{P})(𝔤_{ℓ}) ⊂ 𝔨. Then there is a Levi decomposition

\overline{𝔯}(Q) = 𝔨_{h} ⊕ 𝔤_{ℓ} ⊕ 𝔰

and we let 𝔯(Q) ⊂ 𝔨 denote the opposite parabolic, R(Q) the corresponding parabolic subgroup. We can replace W as 𝔘(𝔨)-module by the quasi-isomorphic complex BGG^{R(Q)}(W). Then there is a canonical quasi-isomorphism

(4.2.25)  \begin{matrix}C_{•}(𝔰, W) & ≅ H_{0}(𝔰, BGG_{•}^{R(Q)}(W)) \\ ⟶^{∼}H_{0}(𝔰, ⨁_{ℓ(s)=•} 𝔘(𝔨) ⊗_{𝔘(𝔯(Q))} E(Λ(W)(s))). \end{matrix}

Here s runs through the set of Kostant representatives relative to R(Q), as in the first part of this section, and we use Λ(W) to designate a highest weight of W.

We now return to the notation E, F, associated to the pair t→t′ ∈ W^{Q}; i.e., associated respectively to the representations E(t′) and E(t) of 𝔨. We extend 𝔘(𝔨) ⊗_{𝔘(𝔯(Q))} E(t)^{∗}(s)) trivially to a module over 𝔘(𝔨 ⊕ 𝔭^{−}) and then restrict to 𝔘(𝔨_{h} ⊕ 𝔭_{h}^{−}) ⊂ 𝔘(F^{0}Lie(P′)). We extend from F^{0}Lie(P′) to F^{0}Lie(P) by letting G_{ℓ} act via the Levi decomposition above. Then as 𝔘(F^{0}Lie(P))-module the right-hand side of the last formula in (4.2.25) is

𝔘(F^{0}Lie(P)) ⊗ 𝔘(𝔯(Q))E(t)^{∗}(s)).

So substituting (4.2.25) into (4.2.24), we obtain

H_{0}(𝔴^{+}, 𝔘(Lie(P))⊗_{𝔘(F^{0}Lie(P))}∧^{d_{P}} (𝔫_{P})^{∗}⊗H_{0}(𝔰, ⨁_{ℓ(s)=•} 𝔘(F^{0}Lie(P))⊗_{𝔘(𝔯(Q))}E(t)^{∗}(s))

which simplifies as

(4.2.26)  H_{0}(𝔴_{P} , 𝔘(Lie(P)) ⊗_{𝔘(𝔯(Q))} E(t)^{∗}(s)) = H_{0}(𝔴_{P} , 𝕄(Λ(t)(s))).

The same calculation, with t replaced by t′, shows that R^{•}π_{+}D_{Z}(F^{∨}) is the automorphic vector bundle associated to

⨁_{ℓ(s)=•} H_{0}(𝔴_{P} , 𝕄(Λ(t′)(s)))

We recognize (4.2.26) as the left-hand side of (4.2.14), where (unfortunately) P and \overline{P} have been interchanged. From here we apply Prop. 4.2.15 (b) to obtain that the correspondence in Lemma 4.2.18 is compatible with the differentials on the two sides of (4.2.4), i.e. that for any s ∈ W \overline{P} ; Q the linearization of the direct image

R^{q}π_{∗}^{(s)}Δ(t, t′) : R^{q}π_{∗}(E)^{s}→R^{q}π_{∗}(F)^{s}

is the homomorphism

D_{M}(R^{d_{P} −q}π_{∗}(F)^{∨})^{s} → D_{M}(R^{d_{P} −q}π_{∗}(E^{∨}))^{s}

associated to the homomorphism 𝕄(Λ(t)(s)) → 𝕄(Λ(t′)(s)) of generalized Verma modules. We conclude:

Theorem (4.2.27). In the situation of (4.1.9), there is a canonical isomorphism of complexes:

R(\overline{π}_{P})_{∗}(i_{R}^{∗})\{BGG(M, Ẽ)\}^{can}⟶^{∼} ⨁_{w∈W^{P}} \{BGG(M_{P} , Ẽ_{μ(h,w)})^{can}⊗C^{•}(Γ_{ℓ,R}, H^{•}(E_{μ(ℓ,w)})[−ℓ(w)]\}.

Here the notation is as in (4.1.10.1).

We adelize the above theorem using the notation Sh_{Σ}, Sh(R), etc., of (3.2). The least awkward expression is in terms of the inverse limit over all toroidal compactifications

S̃h = S̃h(G, X) = \underleftarrow{lim}_{K_{f} ,Σ} Sh_{Σ}

(recall that the level subgroup K_{f} is implicit in the notation). Let i_{R} : S̃h^{R(∗)} ↪ S̃h denote the inclusion of the R(∗)-stratum (inverse limit over all R(∗)-strata at finite level). Let Δ_{1,R,h} denote the projection of the group Δ_{1,R} of (3.2) on G_{h,P} (A_{f}), and let Sh(R)^{+} denote the quotient Sh(G_{h,P} , X(P))/Δ_{1,R,h}. Then R(A_{f}) acts on Sh(R)^{+}, and the adelic version of \overline{π}_{P} is a natural morphism

(4.2.28)  \overline{π}_{R} : S̃h^{R(∗)} → Ind_{R(A_{f})}^{G(A_{f})}S̃h(R)^{+},

where S̃h(R)^{+} is again the projective limit over all toroidal compactifications. For w ∈ W^{R}, let

BGG(Sh(R), w, E) = BGG(Sh(G_{h,P} , X(P)), Ẽ_{μ(h,w)})^{can}⊗C^{•}(X(G_{ℓ,R}), E_{μ(ℓ,w)})[−ℓ(w)]

. Then the formula (3.2.8), with the space H^{•}(w) replaced by the complex BGG(Sh(R), w, E), defines an R(A_{f})-equivariant vector bundle I^{R}(BGG(Sh(R), w, E)) on S̃h(R)^{+}. The adelic version of Theorem (4.2.27) is

Theorem (4.2.29). (a) In the above situation, there is a canonical isomorphism of complexes:

R(\overline{π}_{P})_{∗}(i_{R}^{∗})\{BGG(S̃h(G, X), Ẽ)\}^{can}⟶^{∼}Ind_{R(A_{f})}^{G(A_{f})} ⨁_{w∈W^{R}}\{I^{R}(BGG(Sh(R)), w, E)).

(b) This isomorphism is rational over the reflex field E(G, X).

Proof. Part (a) is just the adelic version of (4.2.27), or rather of its obvious variant for the R-stratum. (Here we are using the fact that the Kostant decomposition for the parabolic subgroup R_{ℓ,P} of G_{ℓ,P} defines a direct sum decomposition in the derived category for the cohomology complex of the R_{ℓ,P} -stratum of the boundary of X(G_{ℓ,P}).) Then (b) is immediate because the identification in Lemma 4.2.18 is rational over E(G, X), as are all subsequent constructions.

(4.2.30) Remark. Upon taking fixed vectors for a level subgroup K_{f}, the above theorem can be given a reasonable expression in terms of toroidal compactifications at finite level.

(4.3) Hodge theory at the boundary, revisited. Let M′_{P} denote a connected component of the mixed Shimura variety associated to a maximal parabolic subgroup P of G, and

(4.3.1)  M′_{P} ⟶^{π_{2}} A_{P} ⟶^{π_{1}} M_{P}

the associated tower of algebraic fibrations (see (1.4.1) or [HZ1, (1.2.5)]). Let, as usual, π (or π_{P}) denote π_{1} ∘π_{2}. For ρ : P → GL(E) a rational representation, there are isomorphisms of local systems on M_{P}:

(4.3.2)  R^{k}π_{∗}Ẽ ≃ H̃^{k}(𝔴_{P} , E).

This follows from the fact that the fibers of π have the homotopy type of the compact nilmanifold Γ_{W_{P}} \W_{P} (ℝ). In fact, (4.3.2) is a consequence of the more basic fact at the cochain level:

(4.3.3)  Rπ_{∗}Ẽ ≃ π_{∗}{C̃^{•}(𝔴_{P} , E)|_{S_{L}}},

where S_{L} is the cross-section of π given by a choice of rational Levi subgroup L ⊂ ^{0}P and basepoint c_{P} (x_{0}) (the latter is fixed by G_{ℓ,P}).

In [HZ2, §5], we saw (among other things) that the above assertions remain true in the Hodge theoretic sense. As J. Wildeshaus has pointed out to us, the arguments in [HZ2] are really assertions about the associated mixed Hodge modules—much of it is already there—and we wish to recast them here.

Let M′_{P,Σ} denote the toroidal partial compactification of M′_{P} determined by Σ; it is not an algebraic variety, as its boundary has infinitely many irreducible components. It contains the boundary stratum that is called ^{<}Z̃_{P,Σ} in [HZ1]. Taking the quotient of the latter by Γ_{ℓ,P} gives rise to the P -stratum ^{<}Z_{P,Σ} in M_{Σ}. And Z_{Σ}(P) is the just the closure of ^{<}Z_{P,Σ} in M_{Σ}. Of course, Z_{Σ}(P) contains Z_{Σ}(R) whenever R ⊆ P, but Z_{Σ}(R) maps surjectively to (M_{P})_{Ξ} under the projection \overline{π} : Z_{Σ}(P) → (M_{P})_{Ξ} (as in (4.1.10.2)) if and only if R is subordinate to P.

We recall some constructions from [HZ2, (5.2)]. We denote by j_{Σ} the inclusion of M′_{P} in M′_{P,Σ}. Let M_{Σ} (called M in [HZ2]) be the unique mixed Hodge module on M′_{P,Σ} with underlying perverse sheaf Rj_{∗}Ẽ, such that its restriction to M′_{P} is the variation of mixed Hodge structure defined by the representation of P on E. Then (π_{∗}M_{Σ}, Rπ_{∗}Ẽ) is a mixed Hodge module on M_{P} with action of Γ_{ℓ,P}. Proposition (4.3.4). [HZ2, (5.2.9),(5.4.19)] In the derived category of mixed Hodge modules, (π_{∗}M_{Σ}, Rπ_{∗}Ẽ) is Γ_{ℓ,P} -equivariantly isomorphic to the complex of variations of mixed Hodge structure π_{∗}{C̃^{•}(𝔴_{P} , E)|_{S_{L}}} with filtrations induced by the mixed Shimura data for P. In particular, the latter is admissible.

Though unusual in general, we expected in this case that the mixed sheaf Rπ_{∗}Ẽ would split as the direct sum of pure sheaves (of different weights). Indeed, the following is an immediate consequence of (4.3.4): Theorem (4.3.5). In the derived category of mixed Hodge modules on M_{P},

Rπ_{∗}Ẽ ≈ H̃^{•}(𝔴, E) ≃ ⨁_{w∈W^{P}} Ẽ_{μ(h,w)} ⊗ E_{μ(ℓ,w)}[−l(w)]

underlies a decomposition of mixed Hodge modules. Each Ẽ_{μ(h,w)}, being determined by an irreducible representation of G_{h,P}, is pure.

Corollary (4.3.6). [HZ2, (5.4.20)] The mixed Hodge structure of H^{i}(M′_{P} , Ẽ) decomposes as the direct sum of mixed Hodge structures:

⨁_{w∈W^{P}} H^{i−l(w)}(M_{P} , Ẽ_{μ(h,w)}) ⊗ E_{μ(ℓ,w)}.

It is time to move on to the boundary cohomology of Z_{Σ} = ∂M_{Σ}. The final result in [HZ2, §5] is the following, which should be compared to (4.3.5) and (4.3.6). It is the Hodge theoretic version of (4.1.9), and, by iteration, it provides the Hodge theoretic version of (4.1.19).

Theorem (4.3.7). [HZ2, (5.6.10)] Let E be a representation of G, so that Ẽ underlies a variation of Hodge structure on M. Let j_{P} : M_{P} → M_{P,Σ(P)} be a suitable toroidal compactification of M_{P}, as in (3.1.2). Suppose that Π(R) = P. Then

i) there is a decomposition in the derived category of mixed Hodge modules on M_{P,Σ(P)}

R\overline{π}_{∗}i_{R}^{∗}Ẽ ≈ ⨁_{w∈W^{P}} R(j_{P})_{∗}Ẽ_{μ(h,w)} ⊗ C^{•}(Γ_{ℓ,R}, E_{μ(ℓ,w)})[−l(w)]

≈ ⨁_{w∈W^{R}} R(j_{P})_{∗}Ẽ_{μ(h,w)} ⊗ C^{•}(Γ_{ℓ,R}^{red}, E_{μ(ℓ,w)})[−l(w)]

≈ R(j_{P})_{∗}C^{•}(Γ_{ℓ,R}, H̃^{•}(𝔴, E));

ii) there is a decomposition of mixed Hodge structures

H_{dn}^{q}(Z_{Σ}(R), Ẽ) ≃ ⨁_{a; w∈W^{P}} H^{a}(M_{P} , Ẽ_{μ(h,w)}) ⊗ H^{q−a−l(w)}(Γ_{ℓ,R}, E_{μ(ℓ,w)}).

Remark. The passage from the sum over W^{P} to the sum over W^{R} is an isomorphism of mixed Hodge modules; cf. Remark (3.1.9.ii)(b).

There are four main ingredients in the proof of the above assertions: the simplicial structure of the boundary (which produces the Γ_{ℓ,R}-cohomology in (ii) above), the calculation on M′_{P} for a single simplex, the irrelevance of the boundary of M_{P}, and the inability of the boundary cohomology here to distinguish the variation of mixed Hodge structure from M′_{P} from the variation of pure Hodge structure on M.

For the second item, one reverts back to M′_{P,Σ}, as the structure of both at the P -stratum is locally the same. Let τ be any cone in Σ_{P}, and i_{τ} : Z_{τ} → M′_{P,Σ} the inclusion. Then, Proposition (4.3.8). There is an isomorphism in the derived category of mixed Hodge modules on M_{P}, given by restriction of that in (4.3.4):

Rπ_{∗}i_{τ}^{∗}Rj_{∗}Ẽ ≈ π_{∗}{C̃^{•}(𝔴_{P} , E)|_{S_{L}}}.

The next proposition, tacit in [HZ2, §5], asserts that a divisor with normal crossings behaves like the boundary of a manifold-with-corners with regard to deleted neighborhood cohomology. (This is consistent with the comparison of nerves in [HZ2, (2.7.8)].)

Proposition (4.3.9). Let D be a divisor with normal crossings, whose irreducible components will be denoted D_{i}, on the complex manifold Y, and let j : Y − D ↪ Y be the inclusion. Let D_{A} = ⋂_{i∈A} D_{i} be an intersection of components of D, and denote by D_{A}^{∘} the set of points of D_{A} that lie on no additional components of D, and ν_{A} : D_{A}^{∘} ↪ D_{A}. Finally, let i_{A} and i_{A}^{∘} denote the respective inclusions of D_{A} and D_{A}^{∘} in Y. Then for any local system L̃ on Y − D,

i_{A}^{∗}Rj_{∗}L̃⟶^{∼}R ν_{A,∗}(i_{A}^{∘})^{∗}Rj_{∗}L̃.

Proof. The assertion is local on D_{A}. Also, there is nothing to prove at points of D_{A}^{∘}. Thus, suppose that y ∉ D_{A}^{∘}, and let D_{B} denote the (non-empty) intersection of the additional divisors passing through y. Then there are local coordinates centered at y such that near y, j : Y −D ↪ Y is (Δ^{∗})^{a} ×(Δ^{∗})^{b} ×Δ^{n−b−a} ⊂ Δ^{n}, where a = #A and b = #B. In these terms, D_{A}^{∘} = {0} × (Δ^{∗})^{b} × Δ^{n−b−a}, and D_{A} = {0} × Δ^{b} × Δ^{n−b−a}. Thus, we may as well assume that n = b + a. Then the stalk of i_{A}^{∗}Rj_{∗}L̃ is C^{•}((Δ^{∗})^{a} × (Δ^{∗})^{b}, L̃), while that of R ν_{A,∗}(i_{A}^{∘})^{∗}Rj_{∗}L̃ is C^{•}((Δ^{∗})^{b}, C^{•}(Δ^{∗})^{a}, L̃). In both cases, one has a complex for computing H^{•}((Δ^{∗})^{a} × (Δ^{∗})^{b}, L̃), and our assertion follows.

In short, one does not have to worry about the compactification of a stratum of the boundary in treating the deleted neighborhood cohomology. This addresses the third item in the outline of the proof of (4.3.7) above.

The fourth and final one is worded imprecisely, and we elaborate now. As mentioned before, Γ_{ℓ,P} \^{<}Z̃_{P,Σ} is analytically isomorphic to ^{<}Z_{P,Σ}; indeed, there is a Γ_{ℓ,P} -invariant deleted neighborhood Õ of ^{<}Z̃_{P,Σ} in M′_{P,Σ} such that Γ_{ℓ,P} \Õ is analytically equivalent to an deleted neighborhood O of ^{<}Z_{P,Σ} in M_{Σ}.

Of course, the local system Ẽ|_{O} underlies a variation of (pure) Hodge structure, viz., the restriction of the one on M. On the other hand, the variation of mixed Hodge structure determined by E as a representation of P is Γ_{ℓ,P} -equivariant, so its restriction to Õ descends to a variation of mixed Hodge structure on O, with the same underlying local system; it is an admissible variation. To distinguish them as mixed Hodge modules, we write Ẽ_{(G)}|_{O} and Ẽ_{(P)}|_{O} resp. The following is an elaboration on [HZ2, (5.6.12)]:

Proposition (4.3.10). Suppose that Π(R) = P. Then the mixed Hodge modules i_{R}^{∗}Ẽ_{(G)}|_{O} and i_{R}^{∗}Ẽ_{(P)}|_{O} are isomorphic.

Proof. We begin by recalling that Ẽ_{(G)}|_{O} and Ẽ_{(P)}|_{O} have the same Hodge filtration, but they differ in their weight filtrations. The latter are determined respectively by the weight homomorphisms w_{G} and w_{P} (as in [HZ1, 1.2.2], where the first one is written h ∘ w). As such, they have the same asymptotic Hodge filtrations, and we want to show that the associated weight filtrations along Z_{Σ}(R) coincide. Let Z_{Σ}^{∘}(R) be the locus of smooth points of Z_{Σ}(R). By (4.3.9) and the uniqueness property [Sa3, 2.11], it is enough to verify that the weights coincide along Z_{Σ}^{∘}(R).

The local monodromy transformations of Ẽ are unipotent. Until we say otherwise, we continue in the setting of an arbitrary admissible variation of mixed Hodge structure (Ṽ, W̃, F) with unipotent local monodromy, defined in a neighborhood of Z_{Σ}^{∘}(R). Then i_{R}^{∗}(Ṽ, W̃, F), can be determined as a mixed Hodge module by iteration on the following construction (iterated deleted neighborhood cohomology, discussed in [HZ2, (3.1.7)]). Let y ∈ Z_{Σ}^{∘}(R); because Π(R) = P, there is a component D of Z_{Σ}(P) passing through y. Also, let i_{D} ↪ M_{Σ} be the inclusion. Then i_{D}^{∗}(Ṽ, W̃, F) is a one-variable degeneration with D as “parameter.”^{8}

The issues that occur are at the level of a filtered local system on the punctured disc Δ^{∗} (the situation transverse to D) with unipotent monodromy. Let N denote the (nilpotent) monodromy logarithm, acting on any reference fiber V of Ṽ. The weight filtration of i_{D}^{∗}(Ṽ, W̃, F) is given by the weight filtration of N relative to W, which is denoted M(N; W). If W is trivial, then M(N; W) is the absolute weight filtration M(N) of N, equal to the convolution of the kernel and image filtrations of N (see [StZ, (2.3)]), which is characterized by the statement

(4.3.11) NM_{k}(N) ⊆ M_{k−2}(N), and N^{k} : Gr_{k}^{M(N)}V → Gr_{−k}^{M(N)}V is an isomorphism.

8The situation in one variable was a basic, pre-existing element in the foundation of Saito’s theory of mixed Hodge modules.

In general, M(N; W) is characterized (if it exists) by

(4.3.12) NM_{k}(N; W) ⊆ M_{k−2}(N; W), and N^{k} : Gr_{k+i}^{M(N;W)}Gr_{i}^{W} V → Gr_{−k+i}^{M(N;W)}Gr_{i}^{W} V is an isomorphism;

in other words, M(N; W) induces M(Gr_{i}^{W} N)[i] on Gr_{i}^{W} V. In total generality, there is no reason that a filtration satisfying (4.3.12) should exist. However, it is one of the conditions defining admissibility that the relative weight filtration exist.

At this point, one appeals to the following fact that we used in [HZ2, (5.6)]:

Proposition (4.3.13). [StZ, (2.14)] Let V be a vector space with increasing filtration W, and N a nilpotent endomorphism of V such that NW_{i} ⊆ W_{i−1}. Then M(N; W) exists if and only if NW_{i} ⊆ W_{i−2}, in which case M(N; W) = W.

We check that this is satisfied in the Shimura variety setting. Let σ be the onedimensional cone in Σ_{P}^{c} that defines D. The monodromy logarithm N around D is given by some non-zero integral element of σ, and this is interior to C_{P}. Thus, its weight filtration is just the one defined by w_{P}. This lowers weights by two, so by (4.3.13), M(N; W) = W, which is also M(N). Thus, there is an isomorphism of mixed Hodge modules:

i_{D}^{∗}Ẽ_{(P)}|_{O} ≃ i_{D}^{∗}Ẽ_{(G)}|_{O}.

Using the fact that i_{R}^{∘} = i_{D} ∘ (i_{R}^{D})^{∘}, where (i_{R}^{D})^{∘} denotes the inclusion of Z_{Σ}^{∘}(R) in D, we get that

i_{R}^{∗}Ẽ_{(P)}|_{O} ≃ i_{R}^{∗}Ẽ_{(G)}|_{O},

on Z_{Σ}^{∘}(R), hence on Z_{Σ}(R), which is what we needed to show.

We conclude by formulating the nerve spectral sequence in the language of this section:

Theorem (4.3.14). (see [HZ2, (5.5.2)])

i) The mixed Hodge module i^{∗}Rj_{∗}Ẽ is canonically isomorphic in the derived category of mixed Hodge modules to the chain complex S_{•}, where

S_{r} = ⨁_{r(R)=r+1} i_{R}^{∗} Rj_{∗}Ẽ

and the differentials are given by restriction. On hypercohomology, the spectral sequence associated to filtration by degree in the latter is the nerve spectral sequence, which is thereby a spectral sequence of mixed Hodge structures. ii) Whenever Π(R) = P, one has, as in (4.3.7, i),

R(\overline{π}_{P})_{∗}i_{R}^{∗}Rj_{∗}Ẽ ≈ R(j_{P})_{∗} C^{•}(Γ_{ℓ,R}, H̃^{•}(𝔴_{P} , E))

≈ ⨁_{w∈W^{P}} R(j_{P})_{∗}Ẽ_{μ(h,w)} ⊗ C^{•}(Γ_{ℓ,R}, E_{μ(ℓ,w)})[−l(w)].

iii) Grading for the Hodge filtration F, one has that Gr_{F}^{p} (i^{∗}Rj_{∗}Ẽ) is canonically quasi-isomorphic to Gr_{F}^{p} S_{•}. The corresponding “nerve” spectral sequence is:

E_{1}^{r,s} = ⨁_{r(R)=r+1} H^{s}(Z_{R}, i_{R}^{∗}Gr_{F}^{p} (Rj_{∗}Ẽ))

≃ ⨁_{r(R)=r+1} H^{s}(Z_{R}, i_{R}^{∗}Gr_{F}^{p} BBG(M, Ẽ)^{can}) ⇒ H^{r+s}(Z, i^{∗}Gr_{F}^{p} BBG(M, Ẽ)^{can}).

iv) Whenever Π(R) = P, one has:

R(\overline{π}_{P})_{∗}i_{R}^{∗}Gr_{F}^{p} BBG(M, Ẽ)^{can} ≈ C^{•}(Γ_{ℓ,R}, \{H^{•}(𝔰𝔩_{2}_{p,R}, Gr_{F}^{p} BBG(M, Ẽ))\}^{can}).

(4.3.15) Remarks. (i) Theorem (4.3.14) has an adelic version, along the lines of Theorem (4.2.29), whose formulation we leave to the reader, noting that it is most easily effected upon replacing the sum over W^{P} by a sum over W^{R}; see the remark following (4.3.7) and the horizontal arrows in (ii), below. We also note that (4.2.29)(b) implies that (4.3.14.i) defines a canonical isomorphism in the derived category of mixed Hodge modules with E(G, X)-rational structure.

(ii) Proposition (3.4.1) also has a version in terms of mixed Hodge modules: if R ⊃ R′ are two proper parabolics of G, with r(R′) = r(R) + 1, then the following diagram commutes:

R(\overline{π}_{R})_{∗}i_{R}^{∗}Rj_{∗}(Ẽ) ⟶^{≈} Ind_{R(A_{f})}^{G(A_{f})} ⨁_{w∈W^{R}}\{I^{R}(R(j_{R})_{∗} C^{•}(X(G_{ℓ,R}), E_{μ(ℓ,w)})[−ℓ(w)])\}
↓ ↓
R(\overline{π}_{R′})_{∗}i_{R′}^{∗}Rj_{∗}(Ẽ) ⟶^{≈} Ind_{R′(A_{f})}^{G(A_{f})} ⨁_{w∈W^{R′}} \{I^{R′}(R(j_{R′})_{∗} C^{•}(X(G_{ℓ,R′}), E_{μ(ℓ,w)})[−ℓ(w)])\}

Here the horizontal arrows are the quasi-isomorphisms of (4.3.14.ii), and the lefthand vertical arrow is defined by the natural restriction. When Π(R) = Π(R′) (resp. Π(R) ≠ Π(R′)) the right-hand vertical arrow is determined as in (3.2.9)(ii) (resp. as in (3.4.1)). We leave the precise formulation to the reader, noting only that in the BGG realization, the quasi-isomorphisms can be replaced by isomorphisms of complexes.

(4.4) The topological cohomology. As usual, let R ∈ P(G). In the “standard” way of doing things, one writes R = L_{R} · W_{R}; then by [Ha],

(4.4.1) H^{•}(\overline{e′(R)}, Ẽ) ≃ H^{•}(e′(R), Ẽ) ≃ H^{•}(Γ_{R}, E) ≃ H^{•}(X(Γ_{L_{R}}), H̃^{•}(𝔴_{R}, E)).

In actuality, the above holds at the cochain level, which can be viewed as an instance of (2.7.8):

(4.4.2)  C^{•}(\overline{e′(R)}, Ẽ) ≈ C^{•}(X(Γ_{L_{R}}), H̃^{•}(𝔴_{R}, E)) ≈ C^{•}(Γ_{L_{R}}, H^{•}(𝔴_{R}, E)).

If we have that Π(R) = P, then G_{h,P} is a direct factor of L_{R}; indeed, we have, as we noted before, that R = G_{h,P} · Q · W_{P}, for Q a parabolic subgroup of G_{ℓ,P}. The group Γ ∩ Q has been denoted Γ_{ℓ,R} (see (3.1.4)). When this is taken into account, we continue as in (3.3) to see that (4.4.2) can be written as:

(4.4.3)  C^{•}(e′(R), Ẽ) ≈ C^{•}(Γ_{h,P} · Γ_{ℓ,R}, H^{•}(𝔴_{P} , E)).

We observe that the only thing on the right-hand side of (4.4.3) that varies when R is, restricted to be subordinate to P, is the “Γ_{ℓ,R}.”

Analogous to (3.5.1), a complex for computing the topological cohomology of the Borel-Serre boundary ∂\overline{X} is the double complex

(4.4.4)  \overline{K}^{r,s} = ⨁_{r(R) = r+1} C^{s}(e′(R), Ẽ),

in which the differential d̃ is the sum of the differential of C^{•} (which increases s by one), and restriction (which increases r by one). We denote the latter again by d_{par}. The nerve spectral sequence in this setting (see [HZ2, (3.5)]) is associated to the filtration R by rank, starting from the analogue of (3.5.2):

(4.4.5)  Gr_{R}^{t}\overline{K}^{•} = ⨁_{r(R) = t+1} C^{•}(e′(R), Ẽ)[−t],

in which d_{par} vanishes. This produces for the E_{1}-term, as in (4.1.15):

E_{1}^{p,q}(\overline{K}^{•}, R) = ⨁_{r(R) = p+1} E_{1}^{p,q}(R), where

(4.4.6) E_{1}^{p,q}(R) = H^{q}(e′(R), Ẽ)

≃ ⨁_{a+j+k = q} H^{a}(M_{P} , H̃^{k}(Γ_{ℓ,R}, H^{j}(𝔴_{P} , E))) by (4.4.3).

When we invoke Kostant’s theorem, this time for the rational parabolic subgroup P of G, with Levi subgroup G_{h,P} × G_{ℓ,P}:

(4.4.7)  H^{•}(𝔴_{P} , E) ≃ ⨁_{w∈W^{P}} (E_{μ(h,w)} ⊗ E_{μ(ℓ,w)}) [−l(w)],

we obtain

(4.4.8) E_{1}^{p,q}(R) ≃ ⨁_{a;w∈W^{P}} H^{a}(M_{P} , Ẽ_{μ(h,w)}) ⊗ H^{q−a−l(w)}(Γ_{ℓ,R}, E_{μ(ℓ,w)})

≃ ⨁_{a;w∈W^{P}} H^{a}(M_{P} , Ẽ_{μ(h,w)}) ⊗ H^{q−a−l(w)}(X(Γ_{ℓ,R}), Ẽ_{μ(ℓ,w)}).

There is some reason to work instead with the filtration R^{h} by holomorphic rank, which is the filtration determined by the decomposition of \overline{M} induced by the natural mapping of \overline{X} onto the Baily-Borel compactification of M (see [Z6, (1.6)(11)]). For this, we have instead

(4.4.9)  i) R_{t}^{h}\overline{K}^{•} = ⨁_{ρ_{h}(R)≤t} C^{•}(e′(R), Ẽ)[1 − r(R)]
ii) Gr_{t}^{R^{h}} \overline{K}^{•} = ⨁_{ρ_{h}(R) = t} C^{•}(e′(R), Ẽ)[1 − r(R)]

= ⨁_{P ∼P_{ρ−t}}C^{•}(Γ_{h,P} , K_{c}^{•}(X(Γ_{ℓ,P}), H^{•}(𝔴_{P} , E))),

with K_{c}^{•} as in (3.5.7), and the differential induced by d + d^{ℓ}.

Remark. We note that the differential d^{h}, viewed as a summand of d_{par}, decomposes canonically as

d^{h} = ∑_{r} d^{h,r},

where, d^{h,r} decreases holomorphic rank by r, and increases R^{ℓ} by r − 1.

¿From (4.4.9, ii), we can now write down a convenient expression for the E_{1}-term of the holomorphic rank spectral sequence (cf. (3.5.8)):

Proposition (4.4.10). In the spectral sequence for R^{h}, the E_{1}-term is

E_{1}^{p,q}(\overline{K}^{•}, R^{h}) = ⨁_{P ∼P_{ρ+p}}Ẽ_{1}^{p,q}(P), where

Ẽ_{1}^{p,q}(P) = ⨁_{a+j+k = p+q} H^{a}(M_{P} , H̃_{c}^{k}(X(Γ_{ℓ,P}), H̃^{j}(𝔴_{P} , E)))

= ⨁_{a+j+k = p+q} H^{a}(Γ_{h,P} , H_{c}^{k}(X(Γ_{ℓ,P}), H̃^{j}(𝔴_{P} , E)))

= ⨁_{a+j+k = p+q} H_{c}^{k}(X(Γ_{ℓ,P}), H̃^{a}(M_{P} , H̃^{j}(𝔴_{P} , E))).

NB—The non-trivial terms in the above are for p ≤ 0.

Proof. We use Kostant’s theorem, again applied to P (4.4.7). This enables us to use the Künneth formula to determine that

(4.4.10.1) Ẽ_{1}^{p,q}(P) = ⨁_{a;w∈W^{P}} H^{a}(M_{P} , Ẽ_{μ(h,w)}) ⊗ H_{c}^{p+q−a−l(w)}(X(Γ_{ℓ,P}), Ẽ_{μ(ℓ,w)}).

We now recombine the terms:

Ẽ_{1}^{p,q}(P) = ⨁_{a;w∈W^{P}} H^{a}(M_{P} , Ẽ_{μ(h,w)} ⊗ H_{c}^{p+q−a−l(w)}(X(Γ_{ℓ,P}), Ẽ_{μ(ℓ,w)}))

= ⨁_{a;w∈W^{P}} H^{a}(M_{P} , H̃_{c}^{p+q−a−l(w)}(X(Γ_{ℓ,P}), Ẽ_{μ(ℓ,w)} ⊗ Ẽ_{μ(h,w)}))

= ⨁_{a+j+k=p+q} H^{a}(M_{P} , H̃_{c}^{k}(X(Γ_{ℓ,P}), H̃^{j}(𝔴_{P} , E))),

which is the first formula for Ẽ_{1}^{p,q}(P) above. The second formula is equivalent, and the third is proved similarly.

We wish to elaborate on the relation between the filtration by holomorphic rank and the Baily-Borel compactification M^{∗} of M = M_{Γ} that was mentioned before (4.4.9). Let f : \overline{M} → M^{∗} be the natural map. Our first observation is that, by degeneration of the relevant spectral sequences, one has the following and its corollary:

Proposition (4.4.11). (R^{i}f_{!}Ẽ)|_{M_{P}} ≃ ⨁_{j+k = i} H̃_{c}^{k}(X(Γ_{ℓ,P}), H̃^{j}(𝔴_{P} , E)).

As we are using a mixture of compact and closed supports, we introduce the symbol H_{ℓ,c}^{•} to refer to that. Then

Corollary (4.4.12). We have the formula for Ẽ_{1}^{p,q}(P) of (4.4.10):

Ẽ_{1}^{p,q}(P) ≃ ⨁_{a+i = p+q} H^{a}(M_{P} , R^{i}f_{!}Ẽ) ≃ H_{ℓ,c}^{p+q}(e′(P), Ẽ).

We consider ∂M^{∗} as a filtered space. Specifically, one has

∂M^{∗} = ⨆_{P maximal} M_{P}

(as a set), and we put for each integer t

(4.4.13)  Y_{t} = ⨆_{hol rk P ≤t} M_{P} .

This defines a finite increasing filtration of ∂M^{∗}, which we also call Y_{∞}, by closed subspaces. Note that Rf_{!}Ẽ is constructible with respect to the stratification of M^{∗} induced from (4.4.13), and we have for all P:

(4.4.14)  ℍ^{•}(M_{P} , Rf_{!}Ẽ) ≃ H_{ℓ,c}^{•}(e′(P), Ẽ).

We now assert:

Proposition (4.4.15). The spectral sequence for the filtration by holomorphic rank (see (4.4.10)) coincides with that of the filtered space ∂M^{∗}, with filtration given in (4.4.13), and with coefficients in F = Rf_{!}Ẽ:

E_{1}^{p,q} = H^{p+q}(Y_{−p}, Y_{−p−1}; F) ⇒ H^{p+q}(Y_{∞}; F).

Proof. Let \overline{Y}_{t} = f^{−1}Y_{t}, a closed subset of ∂\overline{M} that consists of a union of maximal faces. As such, it is topologically a manifold-with-boundary. Note that the boundary of a maximal face \overline{e′(P)} consists of faces \overline{e′(R)} with R ⊂ P; then f(e′(R)) ⊂ M_{P} if and only if R is subordinate to P. It follows that \overline{Y}_{t} is the union of those \overline{e′(R)} for which ρ_{h}(R) ≤ t. From (4.4.9, i), one sees that the cochain complex R_{t}^{h}\overline{K}^{•} is quasi-isomorphic to

C^{•}(\overline{Y}_{t}, ∂\overline{Y}_{t}; Ẽ) ≈ C_{c}^{•}(\overline{Y}_{t} − ∂\overline{Y}_{t}; Ẽ) ≈ C^{•}(Y_{t}; Rf_{!}Ẽ).

This proves our assertion.

The following elementary fact, which is essentially (3.3.4), is fundamental in the determination of the differential d_{1} in the holomorphic rank spectral sequence:

Lemma (4.4.16). Let R be a parabolic subordinate to P, and let

L_{R} = G_{ℓ,R} · G_{h,P}

be the associated decomposition of its Levi subgroup. Let P′ be a maximal parabolic with P′ ≺ P, and put R′ = R ∩ P′. Then the decomposition of L_{R′} is

L_{R′} = G_{ℓ,R′} · G_{h,P′},  with  G_{ℓ,R′} = G_{ℓ,R} · (G_{ℓ,P′} ∩ G_{h,P}).

In particular, when P′ and P are consecutive, G_{ℓ,R′} = G_{ℓ,R}.

When P′ and P are consecutive, we thus have that

(4.4.17)  R′ = L_{R′} · W_{R′} = G_{ℓ,R} · G_{h,P′} · (W_{R} · W_{P′})

is contained in R = G_{ℓ,R} · G_{h,P} · W_{R}. If Γ is sufficiently small, one has the corresponding decomposition of Γ_{R′}. One can elect to factor out W_{P} instead; it is just a question of whether one is viewing R′ as a parabolic subgroup of P or of P′.

At bottom, d^{h} is given by R ↦ R′ = R ∩ P′ whenever Π(R) = P and P′ ≺ P (which entails a shift of rank by one), and the inclusions

(4.4.18)  \overline{e′(P)} ⊃ \overline{e′(R)} ↩ e′(R′) ↪ \overline{e′(P′)},

This is the same situation that was faced in (3.5.11), so d_{1} is the composition of a restriction mapping and a connecting homomorphism. In the topological setting, the most efficient way to formulate this is without the Kostant decompositions, though the latter may be needed for calculations. We obtain the following formula:

Proposition (4.4.19). When P is conjugate to P_{ρ+p} and P′ is conjugate to P_{ρ+p+1} and satisfies P′ ≺ P,

d_{1} : E_{1}^{p,q}(\overline{K}^{•}, R^{h}) → E_{1}^{p+1,q}(\overline{K}^{•}, R^{h})

in the spectral sequence for R^{h}, is the direct sum of maps Ẽ_{1}^{p,q}(P) → Ẽ_{1}^{p+1,q}(P′) given by

H_{ℓ,c}^{i}(e′(P), Ẽ) ⟶^{r} H_{ℓ,c}^{i}(e′(P ∩ P′), Ẽ) ⟶^{δ} H_{ℓ,c}^{i+1}(e′(P′), Ẽ),

where r is restriction and δ a connecting homomorphism.

We quickly compare the nerve and holomorphic rank spectral sequences (see (4.4.6) and (4.4.10) resp.) in the least complicated non-trivial situation, namely when the ℚ-rank of G equals two. In that case, one has standard parabolics, maximal: P_{1} ≻ P_{2}, and minimal P_{12} = P_{1} ∩ P_{2}. Both spectral sequences have only one non-trivial differential, so they both degenerate at E_{2}. For the sake of simplicity of notation, we will omit the symbol for summation over Γ-conjugacy classes in both cases, though it is important not to lose sight of this; said summation is explicit only in our formula for the terms of the holomorphic rank spectral sequence, but it is present in the other one as well.

The respective differentials d_{1} look like:

(4.4.20.N)  \begin{matrix}H^{q}(\overline{e′(P_{1})}, Ẽ) & ⟶^{r} & H^{q}(e′(P_{12}), Ẽ) \\ ⊕ & & || \\ H^{q}(\overline{e′(P_{2})}, Ẽ) & ⟶^{r} & H^{q}(e′(P_{12}), Ẽ) \end{matrix}

(4.4.20.h)  H_{ℓ,c}^{q}(e′(P_{1}), Ẽ) ⟶^{r} H_{ℓ,c}^{q}(e′(P_{12}), Ẽ) ⟶^{δ} H_{ℓ,c}^{q+1}(e′(P_{2}), Ẽ).

The arithmetic quotients for G_{ℓ,P_{1}} and G_{h,P_{2}} are compact; the nilmanifolds coming from the unipotent radicals are always compact. Thus, (4.4.20.h) simplifies to:

(4.4.21.h) H^{q}(e′(P_{1}), Ẽ) ⟶^{r} H^{q}(e′(P_{12}), Ẽ) ⟶^{δ}

H_{c}^{q+1}(e′(P_{2}), Ẽ) ≃ H^{q+1}(\overline{e′(P_{2})}, ∂\overline{e′(P_{2})}; Ẽ).

Since R is finer than R^{h}, we have

(4.4.22)  E_{2}^{0,q}(\overline{K}, R) ↠ E_{2}^{0,q}(\overline{K}, R^{h}),  E_{2}^{1,q}(\overline{K}, R) ↪ E_{2}^{1,q}(\overline{K}, R^{h})

(these relations can also be deduced directly from (4.4.20.N) and (4.4.21.h)). By reason of degree, the spectral sequence degenerates at E_{2} – this need not be true in higher rank – so the total dimension of E_{2} in each degree is the same for both.

(4.5) The mixed Hodge complex filtered by holomorphic rank. Given the isomorphism of deleted neighborhood cohomology for ∂\overline{M} and ∂M_{Σ} provided by [HZ2, (3.5.5)], we begin by replacing \overline{K}^{•} from (4.4.4) by any compatible system of cohomological mixed Hodge complexes (e.g., the de Rham complexes of the corresponding mixed Hodge modules), and put

K̃^{•}(R) = C^{•}(i_{R}^{∗}Ẽ).

We will see that the double complex K̃^{•}, with its Hodge and weight filtrations (one can see F via (4.1) here), is a mixed Hodge complex filtered by R^{h}. The last notion is a technical condition, determined by El Zein, that implies that the spectral sequence for R^{h} is one of mixed Hodge structures. What the latter means is spelled out after [HZ2, (5.5.2)], and it includes the assertion that the differentials of the spectral sequence are morphisms of mixed Hodge structure. A useful criterion for that, which is easy to check in practice, is that R^{h} be a convolutant of the weight filtration W, which in turn implies that R^{h} splits on Gr^{W} K̃^{•} (see [Z3, (3.6)]).

We already know that R is a convolutant of W, with W = R ∗ W^{H}, where W^{H} is the usual Hodge weight filtration on the individual summands of the complex. We also have that R = R^{h} ∗ R^{ℓ} by (3.5.9). However, this fact in general is not enough to give

(4.5.1)  W = R^{h} ∗ (R^{ℓ} ∗ W^{H});

one would need to check that the three filtrations W^{H}, R^{h} and R^{ℓ} form a distributive family in the sense of [Ka, 1.7]. In this case, however, it is evident that (4.5.1) holds, and we leave it to the reader to verify this. We therefore assert:

Proposition (4.5.2). The complex K̃^{•} is a mixed Hodge complex filtered by R^{h} (as well as by R) that determines the mixed Hodge structure of ∂\overline{M} with coefficients in Ẽ.

For emphasis, we restate a consequence of (4.5.2) that was mentioned above:

Corollary (4.5.3). The holomorphic rank spectral sequence in the topological setting is a spectral sequence of mixed Hodge structures.

With (4.5.3) established, we make a Hodge theoretic comparison of the E_{1}-terms of the [topological] nerve and holomorphic rank spectral sequences. For that, we examine the respective formulas, (4.4.8) and (4.4.10.1), for the E_{1}-term. One thing stands out: they both involve the same variations of Hodge structure on M_{P}, viz., those coming from the representation of G_{h,P} on the E_{μ(h,w)}’s. Where they differ is in the finite-dimensional vector spaces (with the trivial Hodge structure) that the cohomology groups H^{a}(M_{P} , Ẽ_{μ(h,w)}) are tensored with. As in [Z2], an “undesirable” term in a formula can disappear because it gets tensored with 0.

Finally, we assert the conclusion towards which we have been heading:

Theorem (4.5.4). The holomorphic rank (resp. nerve) spectral sequence for coherent cohomology abutting to H^{•}(Z_{Σ}, i^{∗}Gr_{F}^{p} {DR(M, Ẽ^{can}}) is canonically isomorphic to the Gr_{F}^{p} of the topological holomorphic rank (resp. nerve) spectral sequence abutting to H^{•}(∂\overline{M}, Ẽ) ≃ H_{dn}^{•}(Z_{Σ}, Ẽ).

(4.5.5) Remark. The same holds for all F^{p}/F^{p′} (with p′ > p).

(4.6) Ghost classes. We set up the question of the existence of ghosts in its natural settings. The reader should expect by now that there are two parallel notions, one for the topological setting (local systems) and one for the coherent setting (vector bundles).

Let Y be a manifold-with-corners. Topologically, it is just a manifold-with-boundary [BS, App.], but the boundary has designated differentiable corner structure. Let {E_{α}} denote the set of closed faces of codimension one in ∂Y, and let Ṽ be a local system on Y. We mention that Y and its interior are homotopically indistinguishable. The faces of ∂Y are themselves manifolds-with-corners. If we define the rank ρ(Y) of Y to be the largest number of maximal (codimension-one) faces of ∂Y having non-empty intersection, or equivalently, the highest codimension of a boundary face, then, a face E of codimension m in ∂Y has rank at most ρ(Y)−m, with equality if and only if E contains a face of highest codimension in ∂Y. Note that this notion of rank is consistent with the intrinsic rank of parabolic subgroups when Y = \overline{M}.

A ghost class in Y with coefficients in Ṽ is an element in H^{•}(Y, Ṽ) with an non-zero image in

(4.6.1)  Gh^{•}(Y, Ṽ) = \frac{ker\{H^{•}(Y, Ṽ) → ⨁_{α} H^{•}(E_{α}, Ṽ)\}}{ker\{H^{•}(Y, Ṽ) → H^{•}(∂Y, Ṽ)\}}

= \frac{ker\{H^{•}(Y, Ṽ) → ⨁_{α} H^{•}(E_{α}, Ṽ)\}}{im\{H^{•}(Y, ∂Y; Ṽ) → H^{•}(Y, Ṽ)\}}

(it is denoted Spect^{•}(Y, Ṽ) in [Z5]). The above is isomorphic to its image under restriction to ∂Y:

(4.6.2) im{H^{•}(Y, Ṽ) → H^{•}(∂Y, Ṽ)} ∩ ker{H^{•}(∂Y, Ṽ) → ⨁_{α} H^{•}(E_{α}, Ṽ)}.

We call Gh^{•}(Y, Ṽ) the ghost group of (Y, Ṽ). Note that the definition depends on the corner structure of Y. In the case of Y = \overline{M}, we have

(4.6.3)  Gh^{•}(\overline{M}, Ẽ) = \frac{ker\{H^{•}(\overline{M}, Ẽ) → ⨁_{P} H^{•}(\overline{e′(P)}, Ẽ)\}}{im\{H^{•}(\overline{M}, ∂\overline{M}; Ẽ) → H^{•}(\overline{M}, Ẽ)\}},

where P runs over maximal parabolics.

Next, let Y be instead a complex manifold, and D an SNC divisor on Y. Let F be a locally-free sheaf on Y. We assume given a partition of the set of irreducible components of D, which decomposes D into a union ⋃_{i} D_{i} (cf. (2.3)). Then a ghost class in Y with coefficients in F, relative to {D_{i}}, is an element of H^{•}(Y, F) that has non-zero image in

(4.6.4)  Gh^{•}(Y, \{D_{i}\}; F) = \frac{ker\{H^{•}(Y, F) → ⨁_{i} H^{•}(D_{i}, F ⊗ O_{D_{i}})\}}{ker\{H^{•}(Y, F) → H^{•}(D, F ⊗ O_{D})\}},

or equivalently: (4.6.5)

im{H^{•}(Y, F) → H^{•}(D, F ⊗O_{D})}∩ ker{H^{•}(D, F ⊗O_{D}) → ⨁_{i} H^{•}(D_{i}, F ⊗O_{D_{i}})}.

Of course, we are most interested in the case where Y = M_{Σ} is a toroidal compactification of a connected component of a Shimura variety, i stands for P, and D_{i} = Z_{Σ}(P); in that case, (4.6.4) becomes: (4.6.6)

Gh^{•}(M_{Σ}, \{Z_{Σ}(P)\}; F) = \frac{ker\{H^{•}(M_{Σ}, F) → ⨁_{P} H^{•}(Z_{Σ}(P), F ⊗ O_{Z_{Σ}(P)})\}}{ker\{H^{•}(M_{Σ}, F) → H^{•}(Z_{Σ}, F ⊗ O_{Z_{Σ}})\}}.

The following is an immediate consequence of what we have developed in (4.1.7) and [HZ2, (5.6)]: Proposition (4.6.7). There is a natural decomposition

Gr_{F}^{•} Gh^{•}(\overline{M}, Ẽ) ≃ Gh^{•}(M_{Σ}, \{Z_{Σ}(P)\}; Gr_{F}^{•} DR(M, Ẽ)^{can}).

(4.6.8) Remark. Contained in the above assertion is the fact that the Hodge components of a ghost class are themselves ghost classes. It follows that the existence of ghost classes for the topological cohomology imply the existence of the same for at least one of the vector bundles on the right-hand side and conversely.

Though we could state the essential content of the Hodge-theoretic criterion (4.6.14) now, we first elect to formulate (4.6.3) in terms of a single filtered mixed Hodge complex. Let

(4.6.9)  \overline{ℭ^{•}} = Cone\{C^{•}(\overline{M}, Ẽ) → C^{•}(∂\overline{M}, Ẽ)\}[−1],

and ℭ̃ its Hodge-theoretic quasi-isomorph:

(4.6.10) ℭ̃^{•} = Cone{C^{•}(M_{Σ}, Rj_{∗}Ẽ) → C_{dn}^{•}(∂M_{Σ}, Ẽ)}[−1].

We have

(4.6.11)  H^{•}(ℭ̃^{•}) ≃ H^{•}(\overline{ℭ^{•}}) ≃ H^{•}(\overline{M}, ∂\overline{M}; Ẽ)).

As the cone of a morphism of mixed Hodge complexes, (4.6.10) is a mixed Hodge complex in the standard way (see [E, p. 76]). We extend the simplicial filtration from the boundary complex by simply setting ℭ̃^{•} = S_{1}, and likewise for \overline{ℭ^{•}}. We now assert:

Proposition (4.6.12). In terms of the filtered mixed Hodge complex (ℭ̃^{•}, S), the ghost group Gh^{•}(\overline{M}, Ẽ) equals

\frac{im\{H^{•}((S_{1}/S_{−1})ℭ̃^{•}) → H^{•}((S_{1}/S_{0})ℭ̃^{•})\}}{im\{H^{•}(S_{1}ℭ̃^{•}) → H^{•}((S_{1}/S_{0})ℭ̃^{•})\}}.

Of course, we may replace ℭ̃^{•} in the above by \overline{ℭ^{•}}.

The “denominator” in (4.6.12) (see (4.6.3)) is, in degree i,

(4.6.13)  im{H_{c}^{i}(M, Ẽ) → H^{i}(M, Ẽ)}.

For E an irreducible representation of G of weight e, the weights occuring in the left-hand side are ≤ i + e, and those occurring in the right-hand side are ≥ i + e. It is immediate, then, that the mixed Hodge structure of (4.6.13) is actually pure of weight i + e. We obtain the following criterion for the existence of ghosts.

Criterion (4.6.14). Assume that Ẽ is pure of weight e. If

im\{H^{i}((S_{1}/S_{−1})ℭ̃^{•}) → H^{i}((S_{1}/S_{0})ℭ̃^{•})\} ≃ ker\{H^{•}(\overline{M}, Ẽ) → ⨁_{P} H^{•}(\overline{e′(P)}, Ẽ)\}

is not pure of weight i + e, then there are ghost classes in H^{i}(\overline{M}, Ẽ).

Remark. Of course, (4.6.14) makes no statement about the possibility that the weight i + e summand of Gh^{i}(\overline{M}, Ẽ) be non-zero. The method of [KR] finds ghosts for Ẽ = ℂ (e = 0) that are in the image of the Borel map (i.e., among the cohomology classes of the invariant forms). These are of weight i. Thus, the above criterion is independent of theirs.

There are other ways to write the ghost group, and each emphasizes a different Hodge-theoretic relation. From the point of view in (4.6.2), we have the equivalent formulas:

(4.6.15) i) Gh^{•}(\overline{M}, Ẽ) ≃ ker\{H^{•}(S_{0}ℭ̃^{•}) → H^{•}(S_{1}ℭ̃^{•}) ⊕ H^{•}((S_{0}/S_{−1})ℭ̃^{•})\}

≃ ker\{H^{•}(∂\overline{M}, Ẽ) ⟶ (H^{•}(\overline{M}, ∂\overline{M}; Ẽ)[1] ⊕ ⨁_{P} H^{•}(e′(P), Ẽ))\}

ii) Gh^{•}(\overline{M}, Ẽ) ≃ ker\{H^{•}(S_{0}ℭ̃^{•}) → H^{•}(S_{1}ℭ̃^{•})\} ∩ im\{H^{•}(S_{−1}ℭ̃^{•}) → H^{•}(S_{0}ℭ̃^{•})\}

≃ ker\{H^{•}(∂\overline{M}, Ẽ) → H^{•}(\overline{M}, ∂\overline{M}; Ẽ)[1]\} ∩ im\{H^{•}(S_{−1}ℭ̃^{•}) → H^{•}(∂\overline{M}, Ẽ)\}.

The apparent difference between (4.6.15) and (4.6.14) is that the former has one looking at (4.6.16)

H^{•}(∂\overline{M}, Ẽ) ⟶ ⨁_{P} H^{•}(e′(P), Ẽ) ≃ ⨁_{P ∼P_{1}} H^{•}(M_{P} , H̃^{•}(𝔴_{P} , E)) ⊕ ⨁_{P ∼P_{2}} H^{•}(𝔴_{P} , E),

whereas (4.6.14) emphasizes

(4.6.17)  H^{•}(\overline{M}, Ẽ) ⟶ ⨁_{P} H^{•}(e′(P), Ẽ).

(4.6.18) Remark. The cohomology H^{i}(S_{−1}ℭ̃^{•}) is that of the codimension-two skeleton of ∂\overline{M}, viz.,

H^{i−1}( ⋃_{r(Q)=2}\overline{e′(Q)}, Ẽ ).

The above union is a non-vacuous disjoint union if and only if G is of ℚ-rank two, in which case it equals

⨁_{r(Q)=2} H^{i−1}(e′(Q), Ẽ).

In effect, the method used in [Z5, App. A] to rule out ghosts for G = GSp(4) (4 × 4 matrices), was based on (4.6.15, ii). The main step was to examine the weight structure of H^{•}(S_{−1}ℭ̃^{•})/im H^{•}(S_{0}/S_{−1}ℭ̃^{•}), viewing H^{•}(S_{−1}ℭ̃^{•}) as the iterated deleted neighborhood cohomology associated to e′(Q) ⊂ \overline{e′(P)} ⊂ M for P of rank one and containing Q. Note that when G = GSp(4), M_{P} is a modular curve (uncompactified) if P ∼ P_{1} and is a point if P ∼ P_{2}.

The local system Ẽ on M gets its weights from the weight homomorphism w_{G}, whereas the local systems on M_{P} get theirs from w_{P}. The two are related by the formula w_{P} = m_{P} ·w_{G} (see [HZ1, (1.2.2.1)]); m_{P} gives rise to the so-called parabolic weights, and the latter are computed from Kostant’s theorem.

4.6.A. Appendix: The case of GSp_{4}. For arithmetic quotients X associated to a reductive group G, with rk_{ℚ}G^{ad} = 2, the ghost group can be approached via (4.6.15, ii) and (4.6.18). It is not so difficult to see what is at stake, and we describe it here, following and correcting [Z5, App. A].

The maximal closed faces of ∂\overline{X} consist of the Borel-Serre compactifications of arithmetic quotients of spaces associated to ℚ-rank one groups. These are of two types, corresponding to the two conjugacy types, P_{1} and P_{2}, of maximal parabolic subgroups of G; the two types of closed boundary face are here denoted \overline{X}_{1} and

115

\overline{X}_{2}, and their respective interiors X_{1} and X_{2}. An intersection of type \overline{X}_{1} ⋂ \overline{X}_{2} is a space of type X_{12}, corresponding to the minimal parabolics P_{12}. The unipotent radicals of the parabolics give rise to proper nilmanifold fibrations, e.g., X_{1} → X̂_{1}, so the cohomology of X_{1} (with coefficients) gets expressed as cohomology with coefficients in Lie algebra cohomology; likewise for X_{2} → X̂_{2} and X_{12} → X̂_{12}. The numbering has been chosen so that when X is a connected Shimura variety, X̂_{1} is a connected Shimura variety of a ℚ-rank one group, and X̂_{2} is collapsed in the map to the Baily-Borel (cf. (4.4)).

The geometry of the boundary is well-known (see the appendix to [HZ2, (3.5)]):

Proposition (4.6.A.1). Any two closed faces of type X_{1} are disjoint, as are two of type X_{2} or X_{12}. The boundary of a face of type X_{1} or X_{2} consists of a finite number of faces of type X_{12}, and each face of type X_{12} is a boundary face of exactly one face of type X_{1}, and of exactly one face of type X_{2}.

Let

(4.6.A.2) P Gh^{i}(X, Ẽ) = coker{⨁ H^{i−1}(X_{1}, Ẽ) ⊕ ⨁ H^{i−1}(X_{2}, Ẽ) ⟶ ⨁ H^{i−1}(X_{12}, Ẽ)},

where the sums are over the set of faces of the given type, be the pre-ghost group; it is only from here that ghost classes could arise. When we decompose the cohomology groups in (4.6.A.2) according to Kostant (cf. (4.6.16)), we get the direct sum of mappings of the form

(4.6.A.3)

i) H^{k}(X̂_{1}, Ẽ_{w^{1}}) → H_{dn}^{k}(X̂_{12}, Ẽ_{w^{1}})
ii) H^{k}(X̂_{2}, Ẽ_{w^{2}}) → H_{dn}^{k}(X̂_{12}, Ẽ_{w^{2}}),

in which w^{j} ∈ W^{j} is a Kostant representative. The mappings (4.6.A.3) give the means for killing potential pre-ghosts.

In the case of GSp(4), everything is at its easiest. The space X̂_{1} and X̂_{2} are both modular for GL(2), and X̂_{12} is just a point, reducing us to k = 0, 1 in (4.6.A.3). It turns out that k = 1 is the more interesting, and we denote by P Gh^{i}(X, Ẽ)_{1} the corresponding subspace of P Gh^{i}(X, Ẽ). (We refer the reader to [Z5] to see that ghost classes cannot come from k = 0.) Since X̂_{1} and X̂_{2} are both non-compact orientable surfaces—call them S—the mappings in (4.6.A.3), when summed over the coundary circles of a single S, can be interpreted as the natural mapping

(4.6.A.4)  H^{1}(S, Ẽ) → H^{0}(∂S^{∗}, R^{1}j_{∗}Ẽ),

where j : S ↪ S^{∗} is the inclusion of S in its compactification with cusps. For simple topological reasons, (4.6.A.4) is surjective if and only if Ẽ is a non-trivial local system. When Ẽ ≃ ℂ, we have instead

(4.6.A.5)  dim coker{H^{1}(S, Ẽ) → H^{0}(∂S^{∗}, R^{1}j_{∗}Ẽ)} = 1.

On the other hand, H^{0}(S, Ẽ) → H^{0}(∂S^{∗}, R^{0}j_{∗}Ẽ) is never surjective.

Let λ_{1} and λ_{2} be the fundamental dominant weights for the C_{2} root system. Let E = E_{λ}, and write λ = m_{1}λ_{1} + m_{2}λ_{2}. Calculation (4.6.A.6). ([Z5, (A.1.16)])

i) E_{w_{1}} is trivial as a representation of SL(2) if and only if l(w^{1}) = 0 or 3, and m_{2} = 0;

ii) Similarly, E_{w_{2}} is trivial as a representation of SL(2) if and only if l(w^{2}) = 0 or 3, and m_{1} = 0.

We rewrite the right-hand side of (4.6.A.3) as

(4.6.A.7)  i) E_{τ_{1}w^{1}},  ii) E_{τ_{2}w^{2}},

where τ_{j} is the non-trivial element of the A_{1} Weyl group W_{j} of P_{j}. This enables us to invoke (4.6.A.1) and write down a basis of P Gh^{•}(X, Ẽ_{λ})_{1}: Proposition (4.6.A.8). Let w^{12} be an element of the Weyl group W^{12} of C_{2}, and E_{w^{12}} the corresponding vector space. Then: P Gh^{•}(X, Ẽ_{λ})_{1} is generated by the E_{w^{12}}, with w^{12} satisfying one of: w^{12} = τ_{1}w^{1}, l(w^{1}) = 0, 3, and m_{2} = 0; or w^{12} = τ_{2}w^{2}, l(w^{2}) = 0, 3, and m_{1} = 0.

It is useful to keep in mind that τ_{1}W^{1} ⋃ τ_{2}W^{2} excludes just the identity element of W^{12}. Similarly, τ_{1}W^{1} ⋂ τ_{2}W^{2} is the longest element (l = 4) of W^{12}. We conclude: Corollary (4.6.A.9). The exceptions to the assertion “P Gh^{i}(X, Ẽ_{λ})_{1} = 0” are given by:

i) i = 2: l(w^{1}) = 0, and m_{2} = 0; or l(w^{2}) = 0, and m_{1} = 0,

ii) i = 5: l(w^{1}) = 3 or l(w^{2}) = 3, and m_{1} = m_{2} = 0.

We prove next: Proposition (4.6.A.10). P Gh^{2}(X, Ẽ_{λ})_{1} ∩ Gh^{2}(X, Ẽ_{λ}) = 0 unless (m_{1}, m_{2}) = (0, 0).

By means of the other calculations in [Z5:(A.1)], we conclude:

Corollary (4.6.A.11). Gh^{2}(X, Ẽ_{λ}) = 0 unless (m_{1}, m_{2}) = (0, 0).

Proof of (4.6.A.10). Let w_{λ} denote the (Hodge) weight of Ẽ_{λ}. It is actually defined only up to an even integer, and the choice will be seen to be irrelevant.

We recall the tables from [Z5:(A.1.15)], in abbreviated form, in which the parabolic weights are determined from Kostant’s theorem:

(4.6.A.10.1)

W^{1}

\underline{ℓ}

parabolic weight highest weight for SL(2)

0  m_{1} + m_{2}  m_{2}
3  −m_{1} − m_{2} − 4  m_{2}

(4.6.A.10.2)

W^{2}

\underline{ℓ}

parabolic weight highest weight for SL(2)

0  m_{1} + 2m_{2}  m_{1}
3  −m_{1} − 2m_{2} − 6  m_{1}

Again, to get a trivial local system at the boundary, we must have either m_{2} = 0 (for 1 ∈ W^{1}) or m_{1} = 0 (for 1 ∈ W^{2}). This gives, for the corresponding trivial local systems on X̂_{1} and X̂_{2}, parabolic weights m_{1} and 2m_{2} resp. It follows that H^{1}(X_{12}, Ẽ_{λ}) is of Hodge theoretic weights w_{λ} − m_{1} + 2 and w_{λ} − 2m_{2} + 2 resp. We must check whether this can match a weight occurring in H^{2}(X, Ẽ_{λ}). The latter are all ≥ 2+w_{λ}, which is higher than those of H^{1}(X_{12}, Ẽ_{λ}) except when m_{1} = m_{2} = 0. Our assertion is proved.

Remark. i) The only possibility for a term E_{w^{12}} entering into P Gh^{i}(X, Ẽ_{λ}) that is not covered by (4.6.A.9) is for w^{12} = 1 (when i = 1). This does not yield a non-zero element of Gh^{i}(X, Ẽ_{λ}), for the weights are wrong ([Z5, (A.1.18)]).

ii) The ghost class constructed in [KR] is for i = 2 and m_{1} = m_{2} = 0.

5. On the comparison of Hodge structures.

It has been more than ten years since the “Zucker Conjecture” (see [Z2, §6]) was proved. We quickly recall the statement in (5.1) below. Let M^{∗} be the Baily-Borel Satake compactification of a connected component M of a Shimura variety.

Theorem (5.1). ([L],[SS]). Let G be the group in the Shimura datum giving rise to M, E a rational representation space for G, and Ẽ the associated local system on M. Then there is a quasi-isomorphism

L_{(2)}^{•}(M^{∗}, Ẽ) ≈ IC^{•}(M^{∗}, Ẽ),

between the sheaves of L^{2} Ẽ-valued differential forms and the Ẽ-valued intersection cochains on M^{∗}.

This has the standard consequence:

Corollary (5.2). For all k, H_{(2)}^{k}(M, Ẽ) ≃ IH^{k}(M^{∗}, Ẽ).

Upon seeing (5.2), it is natural to expect more. Recall that each side of the isomorphism comes with an associated Hodge structure. For the L^{2}-cohomology (the left-hand side), it comes from the L^{2} harmonic forms (see [SZ]); for the intersection cohomology, it comes via Morihiko Saito’s mixed Hodge modules [Sa1]. A priori, the two Hodge structures need not correspond under the isomorphism. However, there is an inevitable conjecture, the Hodge theoretic version of (5.2):

Conjecture (5.3). The isomorphism in (5.2) is an isomorphism of Hodge structures.

This conjecture remains unresolved. A few cases of (5.3), in which M^{∗} has only isolated singular points, are covered in [Z4]. We remind the reader that one cannot even be sure a priori that the Hodge numbers (dim H^{p,q}) coincide.

In the direction of (5.3), we now make the following improvement on [H5, 3.3.9]:

Theorem (5.4). For all k, the mapping

r_{k} : H_{(2)}^{k}(M, Ẽ) ≃ IH^{k}(M^{∗}, Ẽ) → H^{k}(M, Ẽ)

is morphism of mixed Hodge structures.

(5.5) Remarks. i) The image of r_{k} is the lowest non-zero weight level in the mixed Hodge structure of H^{k}(M, Ẽ). This can be deduced from the decomposition theorem: IH^{k}(M^{∗}, Ẽ) is seen to have the same image in H^{k}(M, Ẽ) as IH^{k}(M_{Σ}, Ẽ) has, and the latter determines the lowest non-zero weight level. Thus, (5.4) asserts that the Hodge structures in (5.2) have a given common Hodge-theoretic quotient. By semi-simplicity, this quotient can be embedded as a common substructure.

ii) Let c denote the codimension of the singular locus in M^{∗}. Included in (5.2) is the assertion that r_{k} is an isomorphism whenever k < c, and is injective for k = c. Thus, from (5.4) it follows that (5.3) is true in degrees k ≤ c.

We will be referring to the following commutative diagram:

(5.6)  \begin{matrix}\overline{M} & ⟵^{k} & M & ⟶^{j} & M_{Σ} \\ \overline{p}↓ & & ^{i}↓ & & ↓^{p_{Σ}} \\ M^{∗} & = & M^{∗} & = & M^{∗} \end{matrix}

As in (4.1.8), let DR(M, Ẽ) be the holomorphic de Rham complex of M with values in Ẽ. We have the explicit formula on M_{Σ}:

(5.7)  DR(M, Ẽ)^{can} = Ω_{M_{Σ}}^{•}(log Z_{Σ}) ⊗ E^{can},

a complex quasi-isomorphic to Rj_{∗}Ẽ. We write A^{•}(DR(M, Ẽ)^{can}) to denote the Dolbeault resolution of (5.7); this (or more precisely, its associated single complex) is also quasi-isomorphic to Rj_{∗}Ẽ. We also let

A^{•} = A^{•}(DR(M, Ẽ)^{can})

denote the complex of its global sections. Then H^{•}(A^{•}) = H^{•}(M, Ẽ).

It is clear that A^{•} is a subcomplex of

A_{sia}^{•} = A_{sia}^{•}(M, Ẽ),

the global sections of A_{sia}^{•}(M_{Σ}, Z_{Σ}) ⊗ Ω_{M_{Γ,Σ}}^{•}(log Z_{Γ,Σ}) ⊗ E^{can}, which is equivalently (by (2.6.1)) the global sections of the de Rham complex of forms with moderate growth A_{sia}^{•}(\overline{M}, Ẽ) on \overline{M}. We know that A_{sia}^{•}(\overline{M}, Ẽ) is quasi-isomorphic to Rk_{∗}Ẽ [B2, 7.4]. There is a tautological extension of Ẽ to a local system on \overline{M}_{Γ}, as the latter is a manifold-with-corners, and Rk_{∗}Ẽ is quasi-isomorphic to that. Thus, H^{•}(A_{sia}^{•}) is also H^{•}(M, Ẽ), so the inclusion of A^{•} in A_{sia}^{•} is a quasi-isomorphism.

Both A^{•} and A_{sia}^{•} inherit a filtration F from the usual Hodge filtration on A^{•}(M, Ẽ), which is given in (4.1.2). Note that (A^{•}, F) underlies the standard mixed Hodge complex for H^{•}(M, Ẽ).

Proposition (5.8). The inclusion of filtered complexes (A^{•}, F) ↪ (A_{sia}^{•}, F) is a filtered quasi-isomorphism.

Proof. We must show that for all p,

(5.8.1)  Gr_{F}^{p} A^{•} → Gr_{F}^{p} A_{sia}^{•}

is a quasi-isomorphism. Because of the way the filtration F is given, we can rewrite (5.8.1) as

(5.8.2)  A^{•}{Gr_{F}^{p} (DR(M, Ẽ)^{can})} → A_{sia}^{•}{Gr_{F}^{p} (DR(M, Ẽ)^{can})}.

Now, as we mentioned in §4, Gr_{F}^{p} (DR(M, Ẽ)^{can}) is a complex of O_{M_{Γ,Σ}}-modules. That (5.8.2) is a quasi-isomorphism follows by applying (2.2.5) to each term.^{9} Corollary (5.9). The filtration F on A_{sia}^{•} induces the Hodge filtration on H^{•}(M, Ẽ).

9Alternatively, it is possible to invoke (4.1.12) at this point.

We can now prove (5.4). Let 𝔥^{k} denote the space of Ẽ-valued harmonic k-forms on M. According to [BG], 𝔥^{k} consists of forms of moderate growth. Then, the inclusion

(5.9.1)  𝔥^{•} ↪ A_{sia}^{•}(M, Ẽ),

which is for trivial reasons compatible with F, induces the morphisms r_{k}. We may invoke (5.5, i) and the strictness principle for Hodge structures: a morphism of filtered vector spaces (H, F) → (H′, F′) in which F and F′ define Hodge structures is a morphism of Hodge structures. This completes the proof of (5.4).

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