Plectic Hodge theory I

J. Nekovář and A. J. Scholl

17 April 2017

Introduction

This is the first of a series of preprints providing detailed statements and proofs of the results announced in sections 16–17 of [9]. Here we state and prove the structure theorem for real plectic mixed Hodge structures, describe the extension groups, and show that the singular cohomology of a Hilbert modular variety carries a canonical plectic mixed Hodge structure.

Subsequent papers will deal with plectic Deligne cohomology of Shimura varieties, the plectic polylogarithm sheaf and arithmetic applications.

1 I-filtrations

1.1 I-prefiltrations

(1.1.1) Let I be a locally finite lattice; i.e., I is a poset, and for every i, j ∈ I:

(i) the inf and sup i ∧ j and i ∨ j exist;

(ii) {k ∈ I | i < k < j} is finite.

We view I as a category, with one morphism from i to j whenever i ≤ j.

Let 𝒞 be an abelian category, X ∈ ob 𝒞. Denote by sub X the category of subobjects of X. We assume sub X has arbitrary colimits.

(1.1.2) Definition. An (increasing) I-prefiltration on X is a functor

F: I → sub X.

We usually write F_{i}X instead of F (i). A decreasing I-prefiltration is an I^{op}-prefiltration (usually written F^{i}X).

The I-prefiltered objects (X, F) of 𝒞 form a category preFil_{I}(𝒞). Say that f: (X, F) → (X′, F′) is strict if

∀i ∈ I, f(F_{i}X) = f(X) ∩ F′_{i}X′.

Say that the sequence in preFil_{I}(𝒞)

0 → (X′, F′) ⟶^{f} (X, F) ⟶^{g} (X′′, F′′) → 0

is exact if

(i) the underlying sequence 0 → X′ ⟶^{f} X ⟶^{g} X′′ → 0 is an exact sequence in 𝒞, and

(ii) f and g are strict.

Condition (ii) is equivalent to the exactness of the sequence

0 → F_{i}X′ → F_{i}X → F_{i}X′′ → 0.

for every i ∈ I. This definition makes preFil_{I}(𝒞) into an exact category.

(1.1.3) Let I_{1}, …, I_{r} be locally finite lattices, and I = ∏_{α=1}^{r} I_{α} with the product ordering: \underline{i} ≤ \underline{j} ⟺ (∀α) i_{α} ≤ j_{α}. Suppose that (_{α}F)_{α} are I_{α}-prefiltrations on X. We define an I-prefiltration F = ⋂_{α} _{α}F on X by

F\underline{i}X = ⋂_{α}_{α}F_{i_{α}}X.

(1.1.4) Pedantic note: we are considering filtrations as functors whose values are subobjects, rather than isomorphism classes of subobjects. Therefore certain constructions which follow, involving sums or intersections of subobjects, will only be well-defined up to unique isomorphism.

1.2 I-filtrations

(1.2.1) Definition. (i) A weak I-filtration on X is an I-prefiltration F which commutes with finite limits (i.e. ∀i, j ∈ I, F_{i∧j}X = F_{i}X ∩ F_{j}X).

(ii) An I-filtration on X is an I-prefiltration F such that:

for every finite nonempty subset J = {j_{0}, …, j_{n}} ⊂ I, the sequence

⋯ → ⨁_{α<β<γ} F_{j_{α}∧j_{β}∧j_{γ}}X → ⨁_{α<β} F_{j_{α}∧j_{β}}X → ⨁_{α} F_{j_{α}}X → ∑_{α} F_{j_{α}}X → 0

(1.2.1.1)

is exact.

The maps in (1.2.1.1) are the alternating sums of the inclusions. (Observe that condition (ii) implies that F commutes with finite limits.)

(1.2.2) For J = (j_{0}, …, j_{n}) a finite family in I, let K_{•}(J) = K_{•}(X, F, J) denote the (homological) complex (1.2.1.1), with ∑_{α} F_{j_{α}}X in degree −1. Suppose that j_{0} = j_{1} and J^{∗} = (j_{1}, …, j_{n}). Then K_{•}(J^{∗}) ⊂ K_{•}(J) and the quotient K_{•}(J)/K_{•}(J^{∗}) is easily seen to be acyclic. Hence, if (1.2.1.1) is exact for every finite subset of I, it is exact for every finite family in I. Standard simplicial arguments show also that the condition is independent of the ordering of J.

Every filtration in the usual sense is automatically a ℤ-filtration:

(1.2.3) Proposition. Suppose I is totally ordered. Then any I-prefiltration is an Ifiltration.

Proof. It suffices to check that for any finite filtration (in the usual sense)

0 = F_{−1}X ⊂ F_{0}X ⊂ ⋯ ⊂ F_{n}X = X

the sequence

⋯ → ⨁_{0≤i<j<k≤n} F_{i}X → ⨁_{0≤i<j≤n} F_{i}X → ⨁_{0≤i≤n} F_{i}X → X → 0

is exact. Filtering this complex termwise by F, and applying the spectral sequence of a filtered complex, it is enough to prove it when 0 = F_{m−1}X ⊂ F_{m}X = X for some m. In this case the complex becomes

⋯ → ⨁_{m≤i<j<k≤n} X → ⨁_{m≤i<j≤n} X → ⨁_{m≤i≤n} X → X → 0

which is acyclic (it is the chain complex with coefficients in X for the standard simplex).

(1.2.4) Let I = I_{1} × ⋯ × I_{r} as in (1.1.3). Suppose that (_{α}F)_{α} are weak I_{α}-filtrations on X. Then by definition F = ⋂_{α} _{α}F commutes with finite limits, hence is a weak I-filtration. Conversely:

(1.2.5) Proposition. Let F be a weak I-filtration on X, where I = ∏_{α=1}^{r} I_{α}. Then there exist unique weak I_{α}-filtrations _{α}F such that F = ⋂_{α} _{α}F.

Proof. Simply define

αF_{j} = ∑_{\begin{matrix}\underline{i}=(i_{1},...,i_{r})∈I \\ i_{α}=j \end{matrix}} F\underline{i}.

(1.2.6) In particular, every weak ℤ^{r}-filtration is the intersection of r ℤ-filtrations.

Weak ℤ^{2}-filtrations were considered in [2, X,§2], where they were called bifiltrations. (What is called there an M-filtration coincides with our notion of weak I-filtration.) Moreover one has:

(1.2.7) Proposition. Any weak ℤ^{2}-filtration is a ℤ^{2}-filtration.

Proof. By the above, F = G ∩ H for ℤ-filtrations G, H. Replacing X by the subobject \underrightarrow{lim}_{i} F_{i}X we may assume that X = \underrightarrow{lim}_{m∈ℤ} G_{m}X. Let J = \{j_{α} = (m_{α}, n_{α}) | 0 ≤ α ≤ r\} ⊂ ℤ^{2}

be a finite subset, with m_{0} ≤ m_{1} ⋯ ≤ m_{r}. Consider the modification of the complex (1.2.1.1):

K̃(X, F, J) = [⋯ → ⨁_{α<β<γ} F_{j_{α}∧j_{β}∧j_{γ}}X → ⨁_{α<β} F_{j_{α}∧j_{β}}X → ⨁_{α} F_{j_{α}}X → X]

with X in homological degree −1. For F to be an I-filtration is equivalent to this complex being acyclic in degree ≥ 0. Let G̃ be the truncated filtration on X:

G̃_{p}X = \begin{cases}X & (p > m_{r}) \\ G_{p}X & (m_{0} ≤ p ≤ m_{r}) \\ 0 & (p < m_{0}) \end{cases}

which induces a (finite) filtration on F_{j}X and therefore on the complex K̃(X, F, J). By the spectral sequence for a filtered complex, it is enough to show that for each p, the associated graded grG̃ ; pK̃(X, F, J) is acyclic in degree ≥ 0. For this, compute:

gr_{p}^{G}F_{m,n}X = \begin{cases}0 & (m < p) \\ gr_{p}^{G}H_{n}X & (m ≥ p) \end{cases}

and if α < β then F_{j_{α}∧j_{b}}X = G_{m_{α}}X ∩ H_{n_{α}∧n_{β}}X and so grG̃ ; pK̃(X, F, J) equals, for

m_{0} < p ≤ m_{r},

⋯ → ⨁_{ω≤α<β<γ} gr_{p}^{G}H_{n_{α}∧n_{β}∧n_{γ}}X → ⨁_{ω≤α<β} gr_{p}^{G}H_{n_{α}∧n_{β}}X → ⨁_{ω≤α} gr_{p}^{G}H_{n_{α}}X → X

(1.2.7.1)

where ω is the least α such that m_{α} ≥ p. For p = m_{0} it equals

⋯ → ⨁_{α<β<γ} G_{m_{0}}H_{n_{α}∧n_{β}∧n_{γ}}X → ⨁_{α<β} G_{m_{0}}H_{n_{α}∧n_{β}}X → ⨁_{ω≤α} G_{m_{0}}H_{n_{α}}X → G_{m_{0}}X

(1.2.7.2)

and for p ∉ [m_{0}, m_{r}] it is zero.

Let J′ be the family (n_{α})_{1≤α≤r}. Then (1.2.7.2) is just the complex K̃(G_{m_{0}}X, H, J′) which is acyclic in degree ≥ 0 by (1.2.3). Moreover gr_{p}^{G}H_{n}X = H̄_{n}gr_{p}^{G}X where H̄ is the induced filtration on gr_{p}^{G}X. (We are using [4, (1.1.9)], which states that the two different ways of inducing a filtration on subquotients give the same answer.) So (1.2.7.1) is the complex K̃(gr_{p}^{G}X, H̄, J′_{≥ω}) where J′_{≥ω} = (n_{α})_{ω≤α≤r}, hence by (1.2.3) it is acyclic in degree ≥ 0.

(1.2.8) If r > 2, it is not the case that every weak ℤ^{r}-filtration is a ℤ^{r}-filtration. (The proof above, taking now H to be a weak ℤ^{r−1}-filtration, breaks down at the last step.) The simplest counterexample is: take for X a 2-dimensional vector space Kx ⊕ Ky, and put on it the weak ℤ^{3}-filtration which is the intersection of the ℤ-filtrations F, G, H with F_{0}X = G_{0}X = H_{0}X = 0, F_{2}X = G_{2}X = H_{2}X = X and F_{1}X = Kx, G_{1}X = Ky, H_{1}X = K(x + y).

(1.2.9) Proposition/Definition. The exact category Fil_{I}(𝒞) of I-filtered objects in 𝒞 is closed under finite sums; it is closed under arbitrary sums if they exist in 𝒞.

(1.2.10) Proposition. Let

0 → (X′, F′) → (X, F) → (X′′, F′′) → 0  (1.2.10.1)

be an exact sequence in preFil_{I}(𝒞), and assume that F′′ is an I-filtration. If one of F, F′ is an I-filtration, then so is the other.

Proof. F is an I-filtration on X if and only if K̃(X, F, J)) is acyclic in degree ≥ 0 for all J. Applying the functor K̃(−, −, J) to the exact sequence (1.2.10.1) we obtain an exact sequence of (homological) complexes

0 → K̃(X′, F′, J) → K̃(X, F, J) → K̃(X′′, F′′, J) → 0

By assumption, H_{i}(K̃(X′′, F′′, J)) = 0 from which the result follows.

(1.2.11) Let F be an I-filtration on X, i ∈ I and I_{≤i} = {j ∈ I | j ≤ i}. Then F induces both an I-filtration and an I_{≤i}-filtration on F_{i}X.

On an arbitrary subobject X′ ⊂ X, F induces a weak I-filtration, but in general this need not be an I-filtration.

1.3 Associated graded objects

(1.3.1) Definition. Let X be an object of 𝒞, F an I-prefiltration on X. Define for i ∈ I

F_{<i}X = ∑_{j<i} F_{j}X, gr_{i}^{F} X:= \frac{F_{i}X}{F_{<i}X}

(assuming that the sum exists).

(1.3.2) Definition. An I-prefiltration F on X is bounded below if there exists j ∈ I such that F_{i}X ≠ 0 implies i > j. It is exhaustive if \underrightarrow{lim}_{i∈I} F_{i}X = X. It is finite if it is

bounded below and

(i) there exists k ∈ I such that F_{k}X = X; and (ii) wts(X, F): = {i ∈ I | gr_{i}^{F} X ≠ 0} is finite.

The elements of wts(X, F) will be called the weights of F.

(1.3.3) Proposition/Definition. Let X = ⨁_{i∈I} X(i) be an I-graded object of 𝒞, and set F_{i}X:= ⨁_{j≤i} X(j). Then F is an I-filtration, satisfying gr_{i}^{F} X = X(i). It is finite if and only if J = {i | X(i) ≠ 0} is finite and there exists j ∈ I with j < inf J. An I-filtration of this type is said to be splittable .

Proof. For k ∈ I define F^{(k)} : I → sub X(k) by

F_{i}^{(k)}X(k) = \begin{cases}X(k) & if i ≥ k \\ 0 & otherwise. \end{cases}

Let J = {j_{0}, …, j_{n}} ⊂ I, ordered so that j_{α} ≥ k ⟺ 0 ≤ α ≤ m. Then K(X(k), F^{(k)}, J) is

⋯ → ⨁_{0≤α<β<γ≤m} X(k) → ⨁_{0≤α<β≤m} X(k) → ⨁_{0≤α≤m} X(k) → X(k)

which is exact, hence F^{(k)} is an I-filtration on X(k). As (X, F) = ⨁(X(k), F^{(k)}), F is an I-filtration on X. The rest follows immediately from the definitions.

(1.3.4) Remark. If f = ∑ f_{i}: ⨁ X(i) → ⨁ Y (i) is a graded morphism of I-graded objects, then it is easy to see that the morphism of associated I-filtered objects is strict.

(1.3.5) Proposition. Let

0 → (X′, F′) → (X, F) → (X′′, F′′) → 0

be an exact sequence of I-filtered objects of 𝒞. Then for every i the sequence

0 → gr_{i}^{F′}X′ → gr_{i}^{F} X → gr_{i}^{F′′}X′′ → 0

is exact.

Proof. Let i ∈ I, and let {j_{α}} ⊂ I be any finite subset such that j_{α} < i for all α. We then have a long exact sequence

⋯ → ⨁_{α<β} F_{j_{α}∧j_{β}}X → ⨁_{α} F_{j_{α}}X → F_{i}X → F_{i}X/ ∑_{α} F_{j_{α}}X → 0

and similarly for X′ and X′′. Since 0 → F′_{j}X′′ → F_{j}X → F′_{j}X′ → 0 is exact for every j, we have the corresponding exactness for F_{i}/ ∑_{α} F_{j_{α}}. Passing to the direct limit over all such finite subsets {j_{α}} gives the exactness for gr_{i}^{F}.

For an object Y of 𝒞, let [Y] ∈ K_{0}𝒞 denote the class of Y in the Grothendieck group of 𝒞.

(1.3.6) Theorem. Let F be an I-filtration on X, bounded below and exhaustive.

(i) If F is finite, [X] = ∑_{i}[gr_{i}^{F} X].

(ii) If gr_{i}^{F} X = 0 for every i ∈ I, then X = 0.

(iii) Assume that 𝒞 is semisimple. Then (X, F) is splittable.

Proof. Since F is bounded below, replacing I by a suitable sublattice we may assume that the lattice I has a minimal element.

(i) Let us show by induction that for every i, [F_{i}X] = ∑_{j≤i}[gr_{j}^{F} X]. Let {j_{α}} = {j ∈ I | j < i}, which is finite since I is locally finite and bounded below. By definition of gr^{F} and since F is an I-filtration, we have a (finite) long exact sequence

⋯ → ⨁_{α<β} F_{j_{α}∧j_{β}}X → ⨁_{α} F_{j_{α}}X → F_{i}X → gr_{i}^{F} X → 0  (1.3.6.1)

giving an equality in K_{0}𝒞

[F_{i}X] = [gr_{i}^{F} X] + ∑_{α}[F_{j_{α}}X] − ∑_{α<β}[F_{j_{α}∧j_{β}}X] + · · · .

By induction we may assume that for every j < i, [F_{j}X] = ∑_{j′≤j}[gr_{j′}^{F}X]. Then the equality becomes

[F_{i}X] = [gr_{i}^{F} X] + ∑_{α} ∑_{j≤j_{α}}[gr_{j}^{F} X] − ∑_{α<β} ∑_{j≤j_{α}∧j_{β}}[gr_{j}^{F} X] + · · · .

= [gr_{i}^{F} X] + ∑_{j<i}(#{α | j_{α} ≥ j} − #{(α < β) | j_{α}, j_{β} ≥ j} + …)[gr_{j}^{F} X]

and the parenthesised sum equals 1.

(ii) The exact sequence (1.3.6.1) (which does not require F to be finite) and induction shows that F_{i}X = 0 for all i, and so X = 0 as F is exhaustive.

(iii) Choose for each i a splitting F_{i}X = X(i) ⊕ ∑_{j<i} F_{j}X, so that by induction F_{i}X = ∑_{j≤i} X(j). Write X̃ = ⨁ X(i), F̃_{i}X̃ = ⨁_{j≤i} X(i). Then the obvious map f: (X̃, F̃) → (X, F) is surjective (as F is exhaustive) and is strict. Therefore ker f = (Y, G) is I-filtered, by (1.2.10). Since f is an isomorphism on associated gradeds, by (1.3.5) gr_{i}^{G}Y = 0 for every i and Y = 0, and therefore X = ⨁ X(i).

(1.3.7) Proposition. Let F be a finite weak I-filtration on X. Let Y = F_{j}X for some j ∈ I, with the filtration induced from F. Then

wts(Y, F) = {i ∈ wts(X, F) | i ≤ j}.

Proof. if i ≤ j then obviously gr_{i}^{F} Y = gr_{i}^{F} X. If i ≰ j, then i ∧ j < i, and F_{i}Y = F_{i∧j}Y so gr_{i}^{F} Y = 0.

(1.3.8) From now on, we assume all (pre-)filtrations to be finite.

1.4 Opposed filtrations

(1.4.1) Let F^{•}, G_{•} be decreasing (resp. increasing) I-prefiltrations on X. Consider the I^{opp} × I-prefiltration (well-defined up to unique isomorphism)

F ∩ G: (i, j) ↦ F^{i}X ∩ G_{j}X.

and (gr_{j,F ∩G}^{i}X)_{i,j} the associated graded.

(1.4.2) Definition. F and G are opposed I-filtrations if

(i) F ∩ G is an I^{opp} × I-filtration; and

(ii) i ≠ j ⟹ gr_{j,F ∩G}^{i}X = 0.

The terminology is justified by the following theorem. We assume that either (a) 𝒞 is semisimple, or (b) for every nonzero X in 𝒞, the class [X] ∈ K_{0}𝒞 is nonzero.

(1.4.3) Proposition. (i) Let (F, G) be opposed I-filtrations on X, and set X(i) = F^{i}X ∩ G_{i}X. Then X = ⨁_{i} X(i) and

F^{i}X = ∑_{k≥i} X(k), G_{i}X = ∑_{k≤j} X(k).

In particular, F and G are splittable I-filtrations.

(ii) The category of triples (X, F, G), where (F, G) are opposed I-filtrations on objects X of 𝒞, is equivalent to the category of finitely I-graded objects of 𝒞.

Proof. (i) Since F ∩ G is an I^{opp} × I-filtration, by (1.3.6)

[F^{i} ∩ G_{j}X] = ∑_{\begin{matrix}i′≥i \\ j′≤j \end{matrix}}[gr_{j′,F ∩G}^{i′}X] = ∑_{i≤k≤j}[gr_{k,F ∩G}^{k}X].  (1.4.3.1)

Therefore

[F^{i} ∩ G_{j}X] = 0  if i ≰ j  (1.4.3.2)

and

[F_{i} ∩ G_{i}X] = [gr_{i,F ∩G}^{i}X].

So in case (b), we deduce F^{i} ∩ G_{i}X \underrightarrow{∼} gr_{i,F ∩G}^{i}X. In case (a) we obtain the same isomorphism by splitting the filtration F ∩ G.

Now let J = {j_{1}, …, j_{n}} ⊂ I. The sequence

⨁_{α<β}(F^{j_{α}} ∩ G_{j_{α}})X ∩ (F^{j_{β}} ∩ G_{j_{β}})X → ⨁_{α} X(j_{α}) → X

is exact by condition (ii), and if α ≠ β

(F^{j_{α}} ∩ G_{j_{α}})X ∩ (F^{j_{β}} ∩ G_{j_{β}}) = F^{j_{α}∨j_{β}}X ∩ G_{j_{α}∧j_{β}}X = 0

by (1.4.3.2). Therefore

⨁_{k∈J} X(k) \underrightarrow{∼} ∑_{k∈J} X(k)

and so

⨁_{i≤k≤j} X(k) ⟶(F^{i} ∩ G_{j})X

and by (1.4.3.1) this is an isomorphism. (ii) follows from (i).

2 Plectic Hodge structures

2.1 Pure plectic Hodge structures over ℂ

(2.1.1) All vector spaces in this section will be assumed to be finite-dimensional. We denote the natural generators of ℤ^{r} as \underline{1}_{j} (1 ≤ j ≤ r). If \underline{n} = (n_{j}) ∈ ℤ^{r}, write |\underline{n}| = ∑_{j=1}^{r} n_{j}. If W is a ℤ^{r}-filtration on X, we define the simple filtration sW associated to W by

sW_{w}X = ∑_{|\underline{n}|≤w} W\underline{n}X.

The inclusions W\underline{n}X ⊂ W_{|\underline{n}|}X induce an isomorphism

⨁_{|\underline{n}|=w} grW ; \underline{n} X \underrightarrow{∼} gr_{w}^{sW} X.  (2.1.1.1)

(2.1.2) If F, F̄ are decreasing ℤ^{r}-prefiltrations, and \underline{n} ∈ ℤ^{r}, say that (F, F̄) are \underline{n}- opposed if (F^{•}, F̄^{\underline{n}−•}) are ℤ^{r}-opposed filtrations. (2.1.3) Definition. (i) An unmixed plectic ℂ-Hodge structure is a ℤ^{2r}-graded ℂ-vector space V = ⨁_{\underline{p},\underline{q}∈ℤ^{r}} V^{\underline{p} \underline{q}}. (ii) V is pure of plectic weight \underline{n} ∈ ℤ^{r} if V^{\underline{p} \underline{q}} = 0 unless \underline{p} + \underline{q} = \underline{n}. (iii) V is pure of simple weight w ∈ ℤ if V^{\underline{p} \underline{q}} = 0 unless |\underline{p} + \underline{q}| = w.

(2.1.4) For each \underline{p}, \underline{q} ∈ ℤ^{r} we write ℂ(\underline{p}, \underline{q}) for the unmixed plectic Hodge structure over ℂ of dimension 1 and degree (\underline{p}, \underline{q}).

(2.1.5) Every unmixed plectic ℂ-Hodge structure V has:

• a ℤ-grading V = ⨁_{w∈ℤ} V_{(w)}, where each V_{(w)} is pure of simple weight w ∈ ℤ, and

• a ℤ^{r}-grading V = ⨁_{\underline{n}∈ℤ^{r}} V_{(\underline{n})}, where each V_{(\underline{n})} is pure of plectic weight \underline{n}

and V_{(w)} = ∑_{|\underline{n}|=w} V_{(\underline{n})}. We shall refer to these as the gradings by plectic and simple weight, respectively.

(2.1.6) Let V be an unmixed plectic ℂ-Hodge structure. Define decreasing ℤ-filtrations F_{j}, F̄_{j} (1 ≤ j ≤ r) by

F_{j}^{p}V = ⨁_{p_{j}≥p} V^{\underline{p} \underline{q}}, F̄_{j}^{q}V = ⨁_{q_{j}≥q} V^{\underline{p} \underline{q}}

and ℤ^{r}-filtrations F_{J}, F̄_{J} (for J ⊂ {1, …, r} and J^{c} the complementary subset)

F \underline{a} ; JV = ⋂_{j∉J} F_{j}^{a_{j}}V ∩ ⋂_{j∈J} F̄_{j}^{a_{j}}V = ⨁_{\begin{matrix}p′_{j}≥a_{j} (j∉J) \\ q′_{j}≥a_{j} (j∈J) \end{matrix}} V_{J}^{\underline{p}′\underline{q}′}, F̄_{J} = F_{J^{c}}.

Set F = F_{∅}, F̄ = F̄_{∅}.

(2.1.7) If V is pure of plectic weight \underline{n}, then for every J, (F_{J}, F̄_{J}) are \underline{n}-opposed, since

F \underline{a} ; JV ∩ F̄_{J}^{\underline{n}−\underline{a}}V = V^{\underline{p}′\underline{q}′} with (p′_{j}, q′_{j}) = \begin{cases}(a_{j}, n_{j} − a_{j}) & (j ∉ J) \\ (n_{j} − a_{j}, a_{j}) & (j ∈ J) \end{cases}

(2.1.8) Conversely, if (F, F̄) are \underline{n}-opposed filtrations on V, then by (1.4.3)

V = ⨁_{\underline{p}+\underline{q}=\underline{n}} F \underline{p}V ∩ F̄ \underline{q}V

is a plectic Hodge structure of plectic weight \underline{n}.

(2.1.9) If V is a pure plectic ℂ-Hodge structure of simple weight w ∈ ℤ, then it is still the case that V^{\underline{p} \underline{q}} = F \underline{p}V ∩ F̄ \underline{q}V whenever |\underline{p} + \underline{q}| = w. Conversely, suppose that (F, F̄) are decreasing ℤ^{r}-prefiltrations on a vector space V and that for some w ∈ ℤ,

V = ⨁_{|\underline{p}+\underline{q}|=w} F \underline{p}V ∩ F̄ \underline{q}V.

Then V is a plectic Hodge structure which is pure of simple weight w, and F, F̄ are the associated ℤ^{r}-filtrations.

(2.1.10) The unmixed plectic ℂ-Hodge structures form a ℂ-linear semisimple Tannakian category, which is equivalent to Rep_{ℂ}(𝔾_{m}^{2r}). The plectic weight is given by the character of 𝔾_{m}^{r} ⊂ 𝔾_{m}^{2r}, and the simple weight by the character of 𝔾_{m} diagonally embedded. Its simple objects are the ℂ(\underline{p}, \underline{q}).

2.2 Pure plectic ℝ-Hodge structures

(2.2.1) Let V be an ℝ-vector space, and V_{ℂ} = ⨁ V^{\underline{p} \underline{q}} an unmixed plectic Hodge structure on V_{ℂ} = V ⊗_{ℝ} ℂ. The following conditions are equivalent:

(i) ∀(\underline{p}, \underline{q}), \overline{V^{\underline{p} \underline{q}}} = V^{\underline{q}\underline{p}}.

(ii) ∀j, F̄_{j} = complex conjugate of F_{j}.

(iii) ∀J, F̄_{J} = complex conjugate of F_{J}.

We say that V is a real unmixed plectic Hodge structure (or unmixed plectic ℝ-Hodge structure) if these hold.

(2.2.2) The unmixed plectic ℝ-Hodge structures form an ℝ-linear Tannakian category, equivalent to Rep_{ℝ}(𝕊^{r}), where 𝕊 = R_{ℂ/ℝ}𝔾_{m}. The gradings by plectic and simple weight are given in the same way as in (2.1.10).

(2.2.3) We define an unmixed plectic ℝ-Hodge structure over ℝ to be an unmixed plectic Hodge structure over ℝ, together with r commuting involutions τ_{j} ∈ Aut_{ℝ}(V) satisfying

τ_{j}(F_{i}^{p}) = \begin{cases}F̄_{i}^{p} & if i = j \\ F_{i}^{p} & otherwise. \end{cases}

These form a Tannakian category, equivalent to the category of representations of the group (𝕊⋊ℤ/2ℤ)^{r}, where the nonzero element of ℤ/2ℤ acts on 𝕊 by complex conjugation.

2.3 Mixed structures

(2.3.1) In this and the following section, all vector spaces will be over a field k of characteristic zero. In preparation for the theory of mixed plectic ℝ-Hodge structures, we first review some of Deligne’s theory for usual mixed Hodge structures [4, 5], and discuss its plectic variant.

(2.3.2) Recall from [5] that if V is a finite dimensional vector space over k, the following structures are equivalent:

(a) Three filtrations (F^{•}, F̄^{•}, W_{•}) which are opposed; i.e.,

gr_{n}^{W} F^{p}V ∩ gr_{n}^{W} F̄^{q}V = 0 if p + q = n + 1

and gr_{n}^{W} V = ⨁_{p+q=n} gr_{n}^{W} F^{p}V ∩ gr_{n}^{W} F̄^{q}V

(b) A bigrading V = ⨁ V_{F}^{pq}, plus a nilpotent endomorphism ∂ of V satisfying

∂(V_{F}^{pq}) ⊂ ⨁_{p′<p, q′<q} V_{F}^{p′q′};

(c) A representation ρ: 𝒢 → GL(V), where 𝒢 = 𝔾_{m}^{2} ⋉ U, and U is the pro-unipotent group whose Lie algebra is freely generated by elements ∂^{ab} of bidegree (a, b), a < 0 and b < 0.

(2.3.3) The dictionary (a) ⟹ (b) is given by

F^{p}V = ⨁_{p′≥p} V_{F}^{p′q},  W_{n}V = ⨁_{p+q≤n} V_{F}^{pq}  (2.3.3.1)
F̄^{q} = ⨁_{q′≥q} V pq′ ; F̄,  V pq ; F̄ = (exp ∂)V_{F}^{pq}  (2.3.3.2)

and in the other direction by setting

V_{F}^{pq} = (F^{p}V ∩ W_{p+q}V) ∩ (F̄^{q}V ∩ W_{p+q}V + ∑_{i≥0} F̄^{q−i} ∩ W_{p+q−i−1})  (2.3.3.3)
V pq ; F̄ = (F̄^{q}V ∩ W_{p+q}V) ∩ (F^{p}V ∩ W_{p+q}V + ∑_{i≥0} F^{p−i} ∩ W_{p+q−i−1})  (2.3.3.4)

from which it follows that

V pq ; F̄ ⊂ V_{F}^{pq} + ∑_{p′<p,q′<q} V_{F}^{p′q′}

and that there is a unique endomorphism ∂ as in (b) satisfying (exp ∂)V_{F}^{pq} = V pq ; F̄.

These constructions give rise to equivalences of abelian categories [4, 5], and in particular it follow that a morphism of trifiltered structures (a) is automatically strictly compatible with the filtrations.

(2.3.4) Let G^{•} be the convolution of the filtrations F̄^{•} and W_{•}:

G^{q}V = ∑_{n} F̄^{n+q}V ∩ W_{n}V.

A conceptual way to view the formula (2.3.3.3) is to observe that the filtrations (F^{•}, G^{•}) on the filtered space (V, W_{•}) are 0-opposed: for every p, n ∈ ℤ one has

W_{n}V = (F^{p}V ∩ W_{n}V) ⊕ (G^{−p−1}V ∩ W_{n}V)

giving a splitting of the filtration F^{•}:

F^{p}V ∩ G^{−p}V \underrightarrow{∼} gr_{F}^{p} V.  (2.3.4.1)

Since on gr_{F}^{p} V the filtrations induced by F̄^{•} and W_{•} are (−p)-opposed, they induce a grading on gr_{F}^{p} V, hence via the splitting (2.3.4.1) they induce a bigrading on V, which is the bigrading by the V_{F}^{pq}.

(2.3.5) The equivalence of (b) and (c) is formal, and admits the following simple generalisation:

(2.3.6) Proposition. Let r be a positive integer. The following categories are equivalent:

(i) Rep_{k} 𝒢^{r}.

(ii) The category of triples (V, (V^{\underline{p}\underline{q}}), (∂_{j})), where

V = ⨁_{\underline{p},\underline{q}∈ℤ^{r}} V^{\underline{p} \underline{q}}

is a ℤ^{2r}-graded k-vector space of finite dimension, and (∂_{j})_{1≤j≤r} are commuting endomorphisms of V such that for each j,

∂_{j}(V^{\underline{p} \underline{q}}) ⊂ ∑_{a,b>0} V^{\underline{p}−a\underline{1}_{j},\underline{q}−b\underline{1}_{j}}  (2.3.6.1)

(2.3.7) If V is a representation of 𝒢^{r}, define the plectic Hodge numbers of V to be d^{\underline{p} \underline{q}}(V) = dim V^{\underline{p} \underline{q}} (\underline{p}, \underline{q} ∈ ℤ^{r}).

(2.3.8) The simple objects of Rep_{k} 𝒢^{r} are the 1-dimensional representations V = k(\underline{p}, \underline{q}) (\underline{p}, \underline{q} ∈ ℤ^{r}) with V = V^{\underline{p} \underline{q}} = k, ∂_{j} = 0.

(2.3.9) Write S = {1, …, r}. Let V be a representation of 𝒢^{r}. For each J ⊂ S let ∂_{J} = ∑_{j∈J} ∂_{j}, and define

V_{J}^{\underline{p} \underline{q}}:=(exp ∂_{J})V^{\underline{p} \underline{q}} ⊂ V^{\underline{p} \underline{q}} + ∑_{\begin{matrix}\underline{p}′<\underline{p} \\ \underline{q}′<\underline{q} \end{matrix}} V^{\underline{p}′\underline{q}′}  (2.3.9.1)

giving a family (V_{J}^{∗∗})_{J} of ℤ^{2r}-gradings of V, indexed by subsets J ⊂ S.

Conversely, given gradings V = ⨁_{\underline{p},\underline{q}} V_{J}^{\underline{p} \underline{q}} for all J ⊂ S there exists at most one family

of nilpotent operators (∂_{J})_{J} of negative degree for which V_{J}^{\underline{p} \underline{q}} = (exp ∂_{J})V^{\underline{p} \underline{q}} for every J and (\underline{p}, \underline{q}).

(2.3.10) The representation V of 𝒢^{r} now carries several filtrations:

• The plectic weight filtration W_{•}, a ℤ^{r}-filtration given by

W\underline{n}V = ∑_{\begin{matrix}\underline{p},\underline{q}∈ℤ^{r} \\ \underline{p}+\underline{q}≤\underline{n} \end{matrix}} V_{?}^{\underline{p} \underline{q}}  (\underline{n} ∈ ℤ^{r})  (2.3.10.1)

(where, for each (\underline{p}, \underline{q}), V_{?}^{\underline{p} \underline{q}} can be any of the V_{J}^{\underline{p} \underline{q}}).

• The associated simple weight filtration sW_{•}:

sW_{n}V = ∑_{\begin{matrix}\underline{p},\underline{q}∈ℤ^{r} \\ |\underline{p}+\underline{q}|≤n \end{matrix}} V_{?}^{\underline{p} \underline{q}}  (\underline{n} ∈ ℤ^{r})  (2.3.10.2)

• For each J ⊂ S, a ℤ^{r}-filtration F_{J}^{•} given by

F \underline{a} ; JV = ∑_{\begin{matrix}\underline{p},\underline{q}∈ℤ^{r} \\ p_{j}≥a_{j} (j∉J) \\ q_{j}≥a_{j} (j∈J) \end{matrix}} V_{J}^{\underline{p} \underline{q}}.  (2.3.10.3)

Write F̄_{J}^{•} = F_{J^{c}}^{•}, F^{•} = F_{∅}^{•}.

• For each j ∈ S, ℤ-filtrations F_{j}^{•}, F̄_{j}^{•} given by

F_{j}^{p}V = ∑_{\begin{matrix}\underline{p}′,\underline{q}′∈ℤ^{r} \\ p′_{j}≥p \end{matrix}} V_{J}^{\underline{p}′\underline{q}′}  for any J with j ∉ J

F̄_{j}^{q}V = ∑_{\begin{matrix}\underline{p}′,\underline{q}′∈ℤ^{r} \\ q′_{j}≥q \end{matrix}} V_{J}^{\underline{p}′\underline{q}′}  for any J with j ∈ J.  (2.3.10.4)

(The condition (2.3.6.1) ensures that the first two filtrations are indeed independent of J.) By definition, for every J, W_{•} ∩ F_{J}^{•} is a ℤ^{2r}-filtration on V.

(2.3.11) The filtrations (2.3.10.3) and (2.3.10.4) determine one another by (1.2.5), since

F_{J} = ⋂_{j∉J} F_{j} ∩ ⋂_{j∈J} F̄_{j}.

If r = 1 we simply have F_{1} = F = F_{∅} = F̄_{{1}}, F̄_{1} = F̄ = F_{{1}}.

(2.3.12) If π: U → V is any morphism of representations of 𝒢^{r}, then as π commutes with the ∂_{j}, we have π(U_{J}^{\underline{p} \underline{q}}) ⊂ V_{J}^{\underline{p} \underline{q}}, and therefore π is strictly compatible with all the filtrations. If V = ∑ V_{α} is a sum of subrepresentations then V^{\underline{p} \underline{q}} = ∑ V_{α}^{\underline{p} \underline{q}}, F_{j}^{•}V = ∑ F_{j}^{•}V_{α}.

(2.3.13) The associated graded objects gr_{∗}^{W} V and gr_{∗}^{sW} V are then what we have called (for k = ℂ) unmixed plectic Hodge structures, and they are canonically isomorphic by (2.1.1.1).

(2.3.14) From (2.3.10.3) one sees that for each J ⊂ S, there is an increasing ℤ^{r}- prefiltration G^{J}, complementary to F_{J}, given by

GJ ; \underline{p} = ⨁_{\underline{p}′≱\underline{p}} V_{J}^{\underline{p}′\underline{q}′}.

Write G\underline{p} = G∅ ; \underline{p}. (Note that in general G^{J} is not even a weak ℤ^{r}-filtration.)

(2.3.15) For J ⊂ S and \underline{p}, \underline{q} ∈ ℤ^{r}, let σ_{J}(\underline{p}, \underline{q}) = (\underline{a}, \underline{b}) where

(a_{j}, b_{j}) = \begin{cases}(p_{j}, q_{j}) & if j ∉ J \\ (q_{j}, p_{j}) & if j ∈ J \end{cases}

(2.3.16) Proposition. (i) Let \underline{p} + \underline{q} = \underline{n}. Then

V^{\underline{p} \underline{q}} = (F \underline{p}V ∩ W\underline{n}V) ∩ ((F̄ \underline{q}V ∩ W\underline{n}V) + (G\underline{p}V ∩ W_{<\underline{n}}V)).

(ii) Let J ⊂ S, and (\underline{a}, \underline{b}) = σ_{J}(\underline{p}, \underline{q}). Then

V_{J}^{\underline{p} \underline{q}} = (F \underline{a} ; JV ∩ W\underline{n}V) ∩ ((F \underline{b} ; J^{c}V ∩ W\underline{n}V) + ∑_{\begin{matrix}\underline{a}′+\underline{b}′<\underline{n} \\ \underline{a}′≱\underline{a} \end{matrix}} V^{σ_{J} (\underline{a}′\underline{b}′)}).  (2.3.16.1)

Proof. (i) We have

W_{<\underline{n}}V = (F \underline{p}V ∩ W_{<\underline{n}}V) ⊕ ∑_{\begin{matrix}\underline{p}′+\underline{q}′<\underline{n} \\ \underline{p}′≱\underline{p} \end{matrix}} V^{\underline{p}′\underline{q}′}

and by (2.3.9.1)

F̄ \underline{q}V ∩ W_{<\underline{n}}V = ∑_{\begin{matrix}\underline{p}′+\underline{q}′<\underline{n} \\ \underline{q}′≥\underline{q} \end{matrix}} V_{S}^{\underline{p}′\underline{q}′} ⊂ ∑_{\begin{matrix}\underline{p}′+\underline{q}′<\underline{n} \\ \underline{p}′≱\underline{p} \end{matrix}} V^{\underline{p}′\underline{q}′} = G\underline{p}V ∩ W_{<\underline{n}}V

and

V^{\underline{p} \underline{q}} ⊂ V_{S}^{\underline{p} \underline{q}} + ∑_{\begin{matrix}\underline{p}′+\underline{q}′<\underline{n} \\ \underline{p}′≱\underline{p} \end{matrix}} V^{\underline{p}′\underline{q}′} ⊂ (F̄ \underline{q}V ∩ W\underline{n}V) + ∑_{\begin{matrix}\underline{p}′+\underline{q}′<\underline{n} \\ \underline{p}′≱\underline{p} \end{matrix}} V^{\underline{p}′\underline{q}′}

Recall (cf. [4, (1.2.9)]) a simple lemma from linear algebra:

If A, B, U are subspaces of a vector space V such that U = (A ∩ U) ⊕ (B ∩ U) then (A + U) ∩ (B + U) = (A ∩ B) ⊕ U.

Apply this with U = W_{<\underline{n}}V, A = F \underline{p}V ∩W\underline{n}V, B = (F̄ \underline{q}V ∩W\underline{n}V)+(G\underline{p}V ∩W_{<\underline{n}}V). Then A ∩ U = F \underline{p}V ∩ W_{<\underline{n}} and B ∩ U = G\underline{p} ∩ W_{<\underline{n}}, so by (2.3.14), the conditions are satisfied. Therefore, letting X = A ∩ B be the right-hand side of (i), this gives X ∩ W_{<\underline{n}}V = 0

and (X + W_{<\underline{n}}V)/W_{<\underline{n}}V = (grW ; \underline{n} V)^{\underline{p}\underline{q}}. As V^{\underline{p} \underline{q}} ⊂ X this implies X = V^{\underline{p} \underline{q}}.

(ii) follows by applying the symmetry σ_{J} to (i).

(2.3.17) It follows that from the filtrations ((F_{j})_{j}, W) one may recover the subspaces V_{J}^{\underline{p} \underline{q}}, and hence the action of 𝒢^{S}, inductively by the formula (2.3.16.1). More precisely (cf. [5, (1.1.1)]), the map d_{J} is the automorphism of V given as the composite a_{J}^{−1}a:

V = ⨁ V^{\underline{p} \underline{q}} ∼ ; ⟶ ; a gr_{∗}^{W} V = ⨁(gr_{\underline{p}+\underline{q}}^{W}V)^{\underline{p} \underline{q}} ∼ ; ⟵ ; a_{J} ⨁ V_{J}^{\underline{p} \underline{q}} = V  (2.3.17.1)

They may equally be recovered from the filtrations ((F_{j})_{j}, sW) by the following variant of the previous proposition (whose proof is the same):

(2.3.18) Proposition. (i) Let |\underline{p} + \underline{q}| = w. Then

V^{\underline{p} \underline{q}} = (F \underline{p}V ∩ sW_{w}V) ∩ ((F̄ \underline{q}V ∩ sW_{w}V) + (G\underline{p}V ∩ sW_{w−1}V)).

(ii) Let J ⊂ S, and (\underline{a}, \underline{b}) = σ_{J}(\underline{p}, \underline{q}). Then

V_{J}^{\underline{p} \underline{q}} = (F \underline{a} ; JV ∩ sW_{w}V) ∩ ((F \underline{b} ; J^{c}V ∩ sW_{w}V) + ∑_{\begin{matrix}|\underline{a}′+\underline{b}′|<w \\ \underline{a}′≱\underline{a} \end{matrix}} V^{σ_{J} (\underline{a}′\underline{b}′)}).

(2.3.19) Let V ∈ ob Rep_{k} 𝒢^{r}. Define subspaces

R_{J}(V) = ⨁_{\begin{matrix}p_{i}=q_{i}=0 (i∉J) \\ p_{i}, q_{i}<0 (i∈J) \end{matrix}} V^{\underline{p} \underline{q}}

R(V) = ∑_{J⊂S} R_{J}(V).

Write R_{j}(V) = R_{\{j\}}(V) = ∑_{a,b>0} V^{−a\underline{1}_{j},−b\underline{1}_{j}}. The definitions imply:

(2.3.20) Proposition. R(V) is a subobject of V, and V ↦ R(V) is an exact endofunctor of Rep_{k} 𝒢^{r}.

(2.3.21) Lemma. Assume r > 1. If V = W_{<\underline{0}}V and R(V) = 0, then

Q(V) := ⋂_{i} (F̄_{i}^{0}V + ⋂_{j≠i} F_{j}^{0}V ) = F \underline{0}V ⊕ F̄ \underline{0}V.

Proof. Observe first that Q(V) ⊃ F \underline{0}V +F̄ \underline{0}V, and that F \underline{0}V ∩F̄ \underline{0} = 0 since V = W_{<\underline{0}}V.

Suppose V is unmixed (i.e., the action of 𝒢^{r} factors through 𝔾_{m}^{2r}). Then all the F_{j}^{p}, F̄_{j}^{q} are sums of some of the V^{\underline{p} \underline{q}}, so in the expression for Q(V), intersection is distributive over sum, and therefore

Q(V) = ∑_{J⊂S} (⋂_{j∉J} F̄_{j}^{0}V ∩ ⋂_{i∈J} ⋂_{j≠i} F_{j}^{0}V )

The term J = ∅ equals F̄ \underline{0}. The term J = \{i\} equals

⋂_{j≠i} F_{j}^{0}V ∩ F̄_{j}^{0}V = ∑_{p_{j}, q_{j}≥0 (j≠i)} V^{\underline{p} \underline{q}} = ∑_{p_{j}=q_{j}=0 (j≠i)} V^{\underline{p} \underline{q}}

since V = W_{<\underline{0}}V. This is contained in F \underline{0}V + F̄ \underline{0}V, since R_{\{i\}}(U) = 0. The term J = S equals F \underline{0}V. If 1 < #J < r then ⋂_{i∈J} ⋂_{j≠i} F_{i}^{0}V = F \underline{0}V. So the conclusion holds for V unmixed.

For the general case, consider any nontrivial subobject 0 ≠ U ⊊ V. By the previous Proposition, R(U) = 0 = R(V/U) = 0, so by induction we may assume that the conclusion of the Lemma holds for Q(U) and Q(V/U). Let π: V → V/U be the canonical map. Then as π compatible with all the filtrations (2.3.12), we have π(Q(V)) ⊂ Q(V/U). Moreover

F̄_{i}^{0}V + ∑_{j≠i} F_{j}^{0}V = ∑_{q_{i}≥0} V_{\{i\}}^{\underline{p} \underline{q}} + ∑_{p_{j}≥0 (j≠i)} V_{\{i\}}^{\underline{p} \underline{q}}

and also for U. Since U_{J}^{\underline{p}\underline{q}} = U ∩ V_{J}^{\underline{p} \underline{q}}, we get

U ∩ (F̄_{i}^{0}V + ∑_{j≠i} F_{j}^{0}V) = F̄_{i}^{0}U + ∑_{j≠i} F_{j}^{0}U.

Therefore Q(U) = U ∩ Q(V). We thus have a commutative diagram with exact rows:

0 ⟶ Q(U) ⟶ Q(V) ⟶^{π}  Q(V/U)
∥  ⋃  ∥
0 ⟶ F \underline{0}U ⊕ F̄ \underline{0}U ⟶ F \underline{0}V ⊕ F̄ \underline{0}V ⟶^{π} F \underline{0}(V/U) ⊕ F̄ \underline{0}(V/U) ⟶ 0

and so by the 5-lemma, equality holds in the middle.

(2.3.22) Lemma. Let I, J ⊂ S be disjoint, and define

P_{I,J}(V) = ∑_{i∈I,j∈J} ∑_{p_{i},q_{i},p_{j},q_{j}<0} V^{\underline{p} \underline{q}}.

Then the sum of the inclusions

R_{I}(V) ⊕ R_{J}(V) ⊕ P_{I,J}(V) → V

is injective.

Proof. This follows immediately from the definitions.

2.4 The structure theorem

(2.4.1) In this section we define a category which encapsulates the linear algebra data (2.3.10) arising from representations of 𝒢^{r}. The main theorem (2.4.7) states that this category (which is the plectic analogue of the category of vector spaces with three opposed filtrations) is equivalent to Rep_{k} 𝒢^{r}.

(2.4.2) Define a k-linear category 𝒞_{k}^{r} as follows: its objects are tuples (V, (F_{j}^{•})_{j}, (F̄_{j}^{•})_{j}, W_{•}), where:

• V is a finite-dimension k-vector space.

• F_{j}^{•}, F̄_{j}^{•} (1 ≤ j ≤ r) are decreasing filtrations on V, and W_{•} is an increasing ℤ^{r}-filtration on V.

subject to the condition:

• For each \underline{n} ∈ ℤ^{r}, there is a ℤ^{2r}-grading

grW ; \underline{n} V = ⨁_{\underline{p}+\underline{q}=\underline{n}}(grW ; \underline{n} V)^{\underline{p}\underline{q}}

such that for every J ⊂ S and \underline{a} ∈ ℤ^{r},

grW ; \underline{n} F \underline{a} ; JV = ∑_{\begin{matrix}\underline{p},\underline{q}∈ℤ^{r} \\ p_{j}≥a_{j} (j∉J) \\ q_{j}≥a_{j} (j∈J) \end{matrix}}(grW ; \underline{n} V)^{\underline{p}\underline{q}}.  (2.4.2.1)

where F_{J}^{•} is defined as in (2.3.11).

The morphisms of 𝒞_{k}^{r} are k-linear maps compatible with all of the filtrations.

It is convenient to define a futher category 𝒞_{k}^{r,wk}: the definition is the same as for 𝒞_{k}^{r} except that W is only required to be a weak ℤ^{r}-filtration. It obviously contains 𝒞_{k}^{r} as a full subcategory.

If V is an object of 𝒞_{k}^{r} we define

d^{\underline{p} \underline{q}}(V) = dim(gr_{\underline{p}+\underline{q}}^{W}V)^{\underline{p} \underline{q}}

(2.4.3) Remark. It is not enough in the definition of 𝒞_{k}^{r} to require merely that gr_{•}^{W} F_{j} and gr_{•}^{W} F̄_{j} are given by the grading of gr^{W} V (since the formation of F_{J} from (F_{j}, F̄_{j}) does not a priori commute with passage to gr_{•}^{W}).

(2.4.4) From the definition and (1.2.11), if V is an object of 𝒞_{k}^{r} then so is every W\underline{n}V with the induced filtrations. By (1.3.7), if V ≠ W\underline{n} then the plectic weights of W\underline{n}V form a proper subset of those of V.

(2.4.5) Let V be an object of 𝒞_{k}^{r} and \underline{a}, \underline{b} ∈ ℤ^{r}. Define an object V′ = V (\underline{a}, \underline{b}) of the same category, given by the following data: the underlying vector space is V, and the filtrations are given by

F_{j}^{p}V′ = F_{j}^{p+a_{j}}V, F̄_{j}^{q}V′ = F̄_{j}^{q+b_{j}}V, W\underline{n}V′ = W_{\underline{n}−\underline{a}−\underline{b}}V

Then V ↦ V (\underline{a}, \underline{b}) is an autoequivalence of 𝒞_{k}^{r}.

If J ⊂ S we may define another object μ_{J}V of 𝒞_{k}^{r} as follows: its underlying space is V with the same weight filtrations, and the Hodge filtrations are given by

F_{j}^{p}(μ_{J}V) = \begin{cases}F_{j}^{p}V & (j ∉ J) \\ F̄_{j}^{p}V & (j ∈ J) \end{cases} ,  F̄_{j}^{p}(μ_{J}V) = \begin{cases}F̄_{j}^{p}V & (j ∉ J) \\ F_{j}^{p}V & (j ∈ J) \end{cases} .

(2.4.6) The construction (2.3.10) defines a functor Φ: Rep_{k} 𝒢^{r} → 𝒞_{k}^{r}, which is obviously faithful. We denote the image under Φ of k(\underline{p}, \underline{q}) (2.3.8) by the same symbol. From the definitions, Φ(V ⊗ k(\underline{p}, \underline{q})) = Φ(V)(\underline{p}, \underline{q}), and the dimensions d^{\underline{p} \underline{q}} for V and Φ(V) are the same.

(2.4.7) Theorem. The functor Φ is an equivalence of categories.

(2.4.8) Corollary. (i) Let V be an object of 𝒞_{k}^{r}. Then the weak ℤ^{r}-filtrations F_{J}^{•} are ℤ^{r}-filtrations, the weak ℤ^{2r}-filtrations F_{J}^{•} ∩ W_{•} are ℤ^{2r}-filtrations.

(ii) The category 𝒞_{k}^{r} is abelian, and has tensor product and internal Hom functors, induced by the usual ones on Vec_{k}. The morphisms in 𝒞_{k}^{r} are strictly compatible with all the filtrations in (i).

(2.4.9) First we show that Φ is fully faithful. Let U, V be representations of Rep_{k} 𝒢^{r} and f: Φ(U) → Φ(V) a morphism in 𝒞_{k}^{r}. Consider the decompositions U = ⨁ U_{J}^{\underline{p} \underline{q}}, V = ⨁ V_{J}^{\underline{p} \underline{q}} from (2.3.17). These depend only on the filtrations, and since f is compatible with the filtrations, the definitions imply that f(U_{J}^{\underline{p} \underline{q}}) ⊂ V_{J}^{\underline{p} \underline{q}}. Then the maps (2.3.17.1) fit into a commutative diagram:

U = ⨁ U^{\underline{p} \underline{q}} ∼ ; ⟶ ; a gr_{∗}^{W} U = ⨁(gr_{\underline{p}+\underline{q}}^{W}U)^{\underline{p} \underline{q}} ∼ ; ⟵ ; a_{J} ⨁ U^{\underline{p} \underline{q}} = U
f↓  ^{gr(f)}↓  ^{f}↓
V = ⨁ V^{\underline{p} \underline{q}} ∼ ; ⟶ ; a gr_{∗}^{W} V = ⨁(gr_{\underline{p}+\underline{q}}^{W}V)^{\underline{p} \underline{q}} ∼ ; ⟵ ; a_{J} ⨁ V^{\underline{p} \underline{q}} = V

So f commutes with every d_{J} = a_{J}^{−1}a, and therefore by (2.3.6) f is a morphism in Rep_{k} 𝒢^{r}.

(2.4.10) We now show that Φ is essentially surjective. The proof is in several steps. If V has a single plectic weight — in other words, V = grW ; \underline{n} V for some \underline{n} — then V has the bigrading ⨁_{\underline{p}+\underline{q}=\underline{n}}(grW ; \underline{n} V)^{\underline{p},\underline{q}}, hence comes from a representation of 𝔾_{m}^{r}. The proof will proceed by induction on the number of plectic weights of V.

(2.4.11) The main inductive step is the following: let \underline{n} = \underline{a} + \underline{b}, and V = W\underline{n}V be an object of 𝒞_{k}^{r} such that grW ; \underline{n} V is 1-dimensional of degree (\underline{a}, \underline{b}). Assume that U = W_{<\underline{n}}V is in the essential image of Φ. In paragraphs (2.4.12)–(2.4.18) we will show that V is as well.

(2.4.12) Replacing V by the twist V (−\underline{a}, −\underline{b}) from (2.4.5), we will assume that \underline{a} = \underline{b} = \underline{n} = \underline{0}. For technical reasons we will merely assume that V ∈ 𝒞_{k}^{r,wk}.

(2.4.13) According to the equivalence (2.3.6) we have a decomposition U = ⨁ U^{\underline{p}\underline{q}} together with commuting endomorphisms (∂_{j}^{U}) of U, satisfying (2.3.6.1). To extend the action of 𝒢^{r} from U to V is equivalent to giving a subspace V^{\underline{0}\underline{0}} ⊂ V complementary to U, together with commuting extensions ∂_{j}^{V} of ∂_{j}^{U} to V, satisfying (2.3.6.1).

(2.4.14) Lemma. Suppose R(U) = 0. Then V is isomorphic in 𝒞_{k}^{r,wk} to k(\underline{0}, \underline{0}) ⊕ U, so in particular lies in 𝒞_{k}^{r} and is contained the essential image of Φ.

Proof. Choose elements x_{J} ∈ F \underline{0} ; JV \ F \underline{0} ; JU which have the same image in grW ; \underline{0} V = V/U. Then

x_{J} − x_{I} ∈ U ∩ (F \underline{0} ; IV + F \underline{0} ; JV) ⊂ U ∩ ⋂_{j∉I∪J} F_{j}^{0}V ∩ ⋂_{j∈I∩J} F̄_{j}^{0}V = ⋂_{j∉I∪J} F_{j}^{0}U ∩ ⋂_{j∈I∩J} F̄_{j}^{0}U.

In particular, for any i,

x_{S} − x_{∅} = (x_{S} − x_{{i}}) + (x_{{i}} − x_{∅}) ∈ F̄_{i}^{0}U + ⋂_{j≠i} F_{j}^{0}U

and so by Lemma (2.3.21) x_{S} − x_{∅} ∈ Q(U) = F \underline{0}U ⊕ F̄ \underline{0}U. Applying this instead to μ_{J}V as in (2.4.5) we obtain the relation, for any J ⊂ S:

x_{J} − x_{J^{c}} ∈ F \underline{0} ; JU ⊕ F \underline{0} ; J^{c}U.

Write x_{J} − x_{J^{c}} = y_{J} − y_{J^{c}} where y_{J} ∈ F \underline{0} ; JU. and set x′_{J} = x_{J} − y_{J} = x′_{J^{c}}. We claim

x = x′_{J} is independent of J. Assuming this, then for any J, x ∈ F \underline{a} ; JV if and only if all a_{j} are ≤ 0, and so x defines an isomorphism k(\underline{0}, \underline{0}) ⊕ U ≃ V. As Hom_{𝒞_{r}^{k}}(k(\underline{0}, \underline{0}), U) = 0 (since any morphism is strictly compatible with W) x is unique up to a scalar multiple.

To prove the claim, we have x′_{J} = x′_{J^{c}} ∈ F \underline{0} ; JV ∩ F \underline{0} ; J^{c}V, and

⋂_{j∉J} F_{j}^{0}U ∋ x′_{J} − x′_{∅} = x′_{J^{c}} − x′_{∅} ∈ ⋂_{j∈J} F_{j}^{0}U

and therefore x′_{J} − x′_{∅} ∈ F \underline{0}U. Likewise,

x′_{J} − x′_{S} = x′_{J^{c}} − x′_{S} ∈ ⋂_{j∉J} F̄_{j}^{0}U ∩ ⋃_{j∈J} F̄_{j}^{0}U = F̄ \underline{0}U

and so since x′_{S} = x′_{∅},

x′_{J} − x′_{∅} = x′_{J} − x′_{S} ∈ F \underline{0}U ∩ F̄ \underline{0}U = \{0\}.

(2.4.15) In the general case, since R(U) is a subrepresentation of U, we may consider the space V/R(U) together with the filtrations induced from (F_{j}), W. Assuming the truth of the following lemma, the proof of previous lemma applied to V/R(U) then shows that there exists, up to scalar multiplication, a unique family (x_{J}) ∈ ∏_{J} F \underline{0} ; JV, such that for every I and J, x_{J} − x_{I} ∈ R(U).

(2.4.16) Lemma. V/R(U) together with the induced filtrations is an object of 𝒞_{k}^{r,wk}.

Proof. First we check that the ℤ^{r}-prefiltration on V/R(U) induced from W commutes with finite limits; in other words, that for every \underline{m}, \underline{n} we have (W\underline{m}V + R(U)) ∩ (W\underline{n}V + R(U)) = W_{\underline{m}∧\underline{n}}V ∩R(U). If \underline{m} ≥ 0 or \underline{n} ≥ 0 then as V = W\underline{0}V this is trivial. Otherwise, since W commutes with finite limits and W\underline{0}V = V, we have W\underline{m}V = W_{\underline{m}∧\underline{0}}V ⊂ W_{<\underline{0}}V = U and likewise for \underline{n}, and so we are reduced to the equality W\underline{m}(U/R(U)) ∩ W\underline{n}(U/R(U)) = W_{\underline{m}∧\underline{n}}(U/R(U)), which holds by (2.3.10.1) since U/R(U) ∈ Rep_{k} 𝒢^{r}.

We then need to verify the condition (2.4.2.1). For \underline{n} < \underline{0} it is equivalent to the same condition for U/R(U), which holds since U/R(U) ∈ Rep_{k} 𝒢^{r}. For \underline{n} ≰ \underline{0} we have W\underline{n}(V/R(U)) = W_{\underline{n}∧\underline{0}}(V/R(U)) and \underline{n} ∧ \underline{0} < \underline{n}, hence grW ; \underline{n} (V/R(U)) = 0. It remains to consider \underline{n} = \underline{0}. As grW ; \underline{0} (V/R(U)) is 1-dimensional of degree (\underline{0}, \underline{0}), (2.4.2.1) can be written as

F \underline{a} ; J(V/R(U)) \begin{cases}⊂ U/R(U) & if some a_{j} > 0 \\ ⊄ U/R(U) & otherwise \end{cases}  (2.4.16.1)

Now by the same condition for V, if all a_{j} ≤ 0 then F \underline{a} ; JV ⊄ U, so certainly F \underline{a} ; J(V/R(U)) ⊄ U/R(U). Otherwise suppose say that a_{1} > 0. It is enough to show that F_{1}^{1}V +F̄_{1}^{1}V ⊂ U. But F_{1}^{1}V = F^{(1,−N,...,−N)}V for N sufficiently large, and if this were not in U we would have F^{(1,−N,...,−N)}grW ; \underline{0} V ≠ 0, which is impossible as grW ; \underline{0} V is purely of degree (\underline{0}, \underline{0}).

(2.4.17) Let d_{J}^{U} = exp ∂_{J}^{U} = exp ∑_{j∈J} ∂_{j}^{U}. Extend d_{J}^{U} to an automorphism d_{J}^{V} of V by requiring that x_{J} = d_{J}^{V} (x_{∅}). Let

y_{J} = x_{J} − x_{∅} ∈ R(U) ∩ ⋂_{j∉J} F_{j}^{0}U = R_{J}(U).

In terms of the decomposition V = kx_{∅} ⊕ U, we have matrix representations:

d_{J}^{V} = (\begin{matrix}1 & 0 \\ y_{J} & d_{J}^{U} \end{matrix}) , d_{I}^{V} ∘ d_{J}^{V} = (\begin{matrix}1 & 0 \\ y_{I} + d_{I}^{U}(y_{J}) & d_{I}^{U}d_{J}^{U} \end{matrix}) .

(2.4.18) Suppose I ∩ J = ∅. Then

x_{I∪J} − x_{J} ∈ R(U) ∩ ⋂_{j∈J} F̄_{j}^{0}U ∩ ⋂_{j∉I∪J} F_{j}^{0}U = d_{J}^{U}(R_{I}(U))

so for some ξ ∈ R_{I}(U), x_{I∪J} −x_{J} = d_{J}^{U}ξ. Likewise, for some η ∈ R_{J}(U), x_{I∪J} −x_{I} = d_{I}^{U}η. Now

(d_{J}^{U} − 1)ξ = ∑_{n≥1} \frac{1}{n!}(∂_{J}^{N})^{n}ξ ∈ P_{I,J}(U)

and similarly (d_{I}^{U} − 1)η ∈ P_{I,J}(U). Write

y_{I∪J} = (x_{I∪J} − x_{J}) + y_{J} = ξ + y_{J} + (d_{J}^{U} − 1)ξ
= (x_{I∪J} − x_{I}) + y_{I} = y_{I} + η + (d_{I}^{U} − 1)η.

Then by Lemma (2.3.22) both right hand expressions are decompositions in R_{I}(U) ⊕ R_{J}(U) ⊕ P_{I,J}(U), and so ξ = y_{I}, η = y_{J}, and (d_{J}^{U} − 1)ξ = (d_{I}^{U} − 1)η. Therefore

y_{I∪J} = y_{I} + d_{I}^{U}(y_{J}) = y_{J} + d_{J}^{U}(y_{I})

or equivalently, d_{I∪J}^{V} = d_{I}^{V} ∘d_{J}^{V} = d_{J}^{V} ∘d_{I}^{V}. If we write d_{{j}}^{V} = exp ∂_{j}^{V} then [∂_{{i}}^{V}, ∂_{{j}}^{V}] = 0. Now y_{{j}} ∈ R_{j}(U) and ∂_{j}^{U}(R_{j}(U)) ⊂ R_{j}(U), so since y_{{j}} = (exp ∂_{j}^{V} − 1)(x_{∅}), one has

∂_{j}^{V} (x_{∅}) = \frac{∂_{j}^{U}}{exp ∂_{j}^{U} − 1}(y_{\{j\}}) ∈ R_{J}(U).

Therefore the commuting family of maps (∂_{j}^{V}) satisfies (2.3.6.1), and thus V is in the image of Φ. This completes the proof of the step (2.4.11).

(2.4.19) We now consider a general V ∈ ob 𝒞_{k}^{r}, and proceed by induction on the size of the set wts(V, W) of plectic weights of V. If V has only one plectic weight then V is in the image of Φ. We may therefore assume by induction and (2.4.4) that for every \underline{n} ∈ ℤ^{r} for which W\underline{n}V ≠ V, W\underline{n}V is in the image of Φ. There are two cases:

(i) Suppose that T = wts(V, W) contains s > 1 maximal elements \underline{n}(α) (1 ≤ α ≤ s). Then for every \underline{n} ∈ T, W\underline{n}V is in the essential image of Φ. The complex K̃(V, W, \{\underline{n}(α)\}) of vector spaces (1.2.1.1) is exact since W is a ℤ^{r}-filtration. Let V′ be the cokernel in Rep_{k} 𝒢^{r} of the map

K̃_{2} = ⨁_{α<β} W_{\underline{n}(α)∧\underline{n}(β)}V → ⨁_{α} W_{\underline{n}(α)}V = K̃_{1}.

Then V′ has the same underlying vector space as V. For every \underline{n} with \underline{n} ≤ \underline{n}(α), some α, one has W\underline{n}V′ = W\underline{n}V by construction, so grW ; \underline{n} V′ = grW ; \underline{n} V for all \underline{n} ∈ T. By (2.3.12) the Hodge filtrations on V′ are given by

F_{j}^{•}V′ = ∑_{α} F_{j}^{•}W_{\underline{n}(α)}V ⊂ F_{j}V.

Therefore (grW ; \underline{n} V′)^{\underline{p} \underline{q}} ⊂ (grW ; \underline{n} V)^{\underline{p} \underline{q}} for every (\underline{p}, \underline{q}), and so equality holds. Therefore F_{j}^{•}V = F_{j}^{•}V′ and hence V = V′ is in the essential image of Φ.

(ii) Otherwise, V has a unique maximal plectic weight \underline{n}, so V = W\underline{n}V and grW ; \underline{n} V ≠ 0, and by induction U = W_{<\underline{n}}V is in the essential image of Φ. Let \{x_{α} | 1 ≤ α ≤ N\} be a basis for grW ; \underline{n} V adapted to the grading (2.4.2.1). Let π: V → grW ; \underline{n} V be the quotient, and V_{α} = π^{−1}(kx_{α}). Then with the induced filtrations, V_{α} is in 𝒞_{k}^{r} and satisfies the hypotheses of (2.4.11), so is in the essential image of Φ. The kernel of the vector space surjection ⨁ V_{α} → V is the sum-zero subspace (U^{N})^{Σ=0}. Then taking V′ to be the cokernel in Rep_{k} 𝒢^{r} of (U^{N})^{Σ=0} → ⨁ V_{α} the same argument as in (i) shows that V = ∑ V_{α} is also in the essential image of Φ.

2.5 Real mixed plectic Hodge structures

(2.5.1) We define a mixed plectic ℂ-Hodge structure to be an object of any of the equivalent categories in Theorem (2.4.7), with k = ℂ.

(2.5.2) We may then define the categories of mixed plectic ℝ-Hodge structures and mixed plectic ℝ-Hodge structures over ℝ in exactly the same way as (2.2). Specifically, a mixed plectic ℝ-Hodge structure is a finite-dimensional real vector space V, together with

(i) decreasing ℤ-filtrations F_{j}^{•} (1 ≤ j ≤ r) on V_{ℂ} = V ⊗_{ℝ} ℂ; and

(ii) a ℤ^{r}-filtration W on V

such that (V_{ℂ}, (F_{j}), (F̄_{j}), W) is a mixed plectic ℂ-Hodge structure. A mixed plectic ℝ-Hodge structure over ℝ is a mixed plectic ℝ-Hodge structure (V, (F_{j}), W) together with r commuting involutions τ_{j} ∈ Aut_{ℝ}(V) satisfying

τ_{j}(F_{i}^{p}) = \begin{cases}F̄_{i}^{p} & if i = j \\ F_{i}^{p} & otherwise. \end{cases}

(2.5.3) These objects form ℝ-linear Tannakian categories ℝ–HS_{r}, ℝ–HS_{r}^{+} respectively (with the obvious morphisms). They are equivalent to the categories of real representations of the groups 𝒢_{ℝ}^{r} and (𝒢_{ℝ} ⋊ ℤ/2ℤ)^{r} respectively, where 𝒢_{ℝ} is the real form of 𝒢 which is denoted 𝔐 in [5, §2].

(2.5.4) There are plectic Tate Hodge structures ℝ(\underline{n}) (\underline{n} ∈ ℤ^{r}) in both categories. The underlying vector space is (2πi)^{|\underline{n}|}ℝ, and the plectic Hodge structure is the unique one for which

ℝ(\underline{n})_{ℂ} = (gr_{−2\underline{n}}^{W}ℝ(\underline{n}))^{−\underline{n},−\underline{n}}.

In ℝ–HS_{r}^{+}, the real Frobenius τ_{j} acts as (−1)^{n_{j}}.

In the next section we compute the Ext-groups in these categories.

3 Extensions

3.1 Extensions in the category 𝒞_{k}^{r}

(3.1.1) Let (V, (F_{j})_{j}, (F̄_{j})_{j}, W) be an object in 𝒞_{k}^{r}. Let 1 be the unit object of 𝒞_{k}^{r}. We will give an explicit complex computing R Hom_{𝒞_{k}^{r}}(1, V).

(3.1.2) We will need the results and methods of [1]. In particular, we start from the more-or-less well-known quasi-isomorphism (in the derived category of k-vector spaces) for the case r = 1:

R Hom_{𝒞_{k}^{1}}(1, V) \underrightarrow{∼} [F^{0}W_{0}V ⊕ F̄^{0}W_{0}V ⟶^{(x,y)↦x−y} W_{0}V ] .  (3.1.2.1)

(3.1.3) In [1] this is proved for ℝ-Hodge structures, and the formulation is slightly different. As we will use the same argument below for general r, we spell it out here.

(3.1.4) Let Γ̃(V) = [Γ̃^{0}(V) → Γ̃^{1}(V)] be the complex on the right-hand side of (3.1.2.1). The functors Γ̃^{i}: 𝒞_{k}^{1} → Vec_{k} are exact, since the morphisms in 𝒞_{k}^{1} are strictly compatible with the filtrations. There is a canonical isomorphism of functors

Hom_{𝒞_{k}^{1}}(1, −) \underrightarrow{∼} ker(d: Γ̃^{0} → Γ̃^{1})  (3.1.4.1)

which takes f: 1 = k(0, 0) → V to (f(1), f(1)) ∈ ker(d).

(3.1.5) Extend Γ̃ to a functor C(𝒞_{k}^{1}) → C(Vec_{k}), where C(−) is the category of chain complexes. It is easily seen to have the following properties:

(i) Γ̃ takes quasi-isomorphisms to quasi-isomorphisms.

(ii) If K^{•} ∈ C(𝒞_{k}^{1}) is bounded below (resp. above) then so is Γ̃(K^{•}).

Therefore it descends to a functor Γ̄: D(𝒞_{k}^{1}) → D(Vec_{k}), which preserves boundedness conditions. The isomorphism (3.1.4.1) gives by derivation a map

R Hom_{𝒞_{k}^{1}}(1, −) → Γ̄(−).

This is an isomorphism of functors. To see this, it is enough to know that for an object V ∈ 𝒞_{k}^{1}, it induces an isomorphism on H^{i}. We have already seen this for H^{0}, and the usual argument for Hodge structures (cf. [1, Lemma 1.8]) shows that it is an isomorphism on H^{1}. Since 𝒞_{k}^{1} = Rep_{k} 𝒢 and 𝒢 is the semidirect product of 𝔾_{m} and a free prounipotent group, H^{i} = 0 for i > 1.

(3.1.6) Define for any I ⊂ S = {1, …, r} and any J ⊂ I the space

Γ̃_{I,J}(V) = ⋂_{j∈I\J} F_{j}^{0}W\underline{0}V ∩ ⋂_{j∈J} F̄_{j}^{0}W\underline{0}V

and set Γ̃_{I}(V) = ⨁_{J⊂I} Γ_{I,J}(V). These are functors 𝒞̃_{k}^{r} → Vec_{k}, and since the morphisms in 𝒞_{k}^{r} are strictly compatible with the filtrations, they are exact.

(3.1.7) We may then define a multidimensional complex indexed by {0, 1}^{r}, with Γ̃_{I}(V) placed in degree (ε_{1}, …, ε_{r}) where I = {j | ε_{j} = 0}. The differentials in the i-direction are

d_{i}: Γ̃_{I}(V) → Γ̃_{I\{i}}(V), (x_{J})_{J⊂I} ↦ (y_{J})_{J⊂I\{i}}

where y_{J} = x_{J} − x_{J∪{i}}.

(3.1.8) Let Γ̃_{•}(V) be the associated simple complex. It is a functor from 𝒞_{k}^{r} to C(Vec_{k}), the category of chain complexes of k-vector spaces, and so extends to a functor Γ̃: C(𝒞_{k}^{r}) → C(Vec_{k}). If r = 1, this agrees with the functor considered in (3.1.4).

(3.1.9) Lemma. (i) There is a functorial isomorphism Hom_{𝒞_{k}^{r}}(1, V) → H^{0}(Γ̃_{•}(V)).

(ii) Γ̃ takes quasi-isomorphisms to quasi-isomorphisms.

Proof. (i) In degrees 0 and 1, Γ̃_{•}(V) is

⨁_{J} F \underline{0} ; JW\underline{0}V → ⨁_{i∈S} ⨁_{J⊂S\\{i\}} [ ⋂_{j∉J∪\{i\}} F_{j}^{0}W\underline{0}V ∩ ⋂_{j∈J} F̄_{j}^{p}W\underline{0}V ]

which shows that H^{0}(Γ̃_{•}(V)) = ⋂_{j} F_{j}^{0}V ∩ F̄_{j}^{0}V ∩ W\underline{0}V = Hom(1, V).

(ii) Taking cones, it is enough to check that if K^{•} ∈ C(𝒞_{k}^{r}) is acyclic, then so is Γ̃(K^{•}). This holds since every F_{J} ∩ W is a ℤ^{2r}-filtration with respect to which morphisms in 𝒞_{k}^{r} are strict (2.4.8).

(3.1.10) By (ii) and the exactness of Γ̃_{I}, just as in the case r = 1, the functor Γ̃ descends to a functor Γ̄ on derived categories, preserving boundedness conditions.

(3.1.11) Theorem. The map of functors R Hom_{𝒞_{k}^{r}}(1, −) → Γ̄ induced by (3.1.9)(i) is an isomorphism.

Proof. It is enough to show that for any simple object of 𝒞_{k}^{r}, the map induces isomorphisms on H^{∗}. Consider the external tensor product ⊠: 𝒞_{k}^{1} × ⋯ × 𝒞_{k}^{1} → 𝒞_{k}^{r}. The simple object k(\underline{p}, \underline{q}) is isomorphic to ⊠k(p_{i}, q_{i}). Let us prove the isomorphism of H^{∗} for V = ⊠V_{i}, for any V_{i} ∈ 𝒞_{k}^{1} (1 ≤ i ≤ r). But Γ̃(V) is then simply the tensor product of the complexes Γ̃(V_{i}), and by the Künneth formula for cohomology of 𝒢^{r},

R Hom_{𝒞_{k}^{r}}(1, ⊠V_{i}) = ⨂ R Hom_{𝒞_{k}^{1}}(1, V_{i}).

So the result follows from (3.1.2.1).

3.2 Extensions of mixed plectic Hodge structure

(3.2.1) Let V be a mixed plectic ℝ-Hodge structure. Then the filtrations F_{j}, F̄_{j} are complex conjugate, and W is real. Therefore we may define a complex conjugation c on the complex Γ̃(V_{ℂ}) by:

c: Γ̃_{I,J}(V_{ℂ}) → Γ̃_{I,I\J}(V_{ℂ}), x ↦ (−1)^{r−#I} x̄

and set

Γ̃_{ℝ}(V) = Γ̃(V_{ℂ})^{c=1}.

If r = 1 then Γ̃_{ℝ}(V) = [F^{0}W_{0}V_{ℂ} ⟶^{d} W_{0}V (1)], with d given by (twice) the projection V_{ℂ} = V ⊕ V (1) → V (1) onto the imaginary part.

As passing to c-invariants is exact, we obtain a functor Γ̄_{ℝ} on the derived category, and one obtains:

(3.2.2) Corollary. There is a canonical isomorphism of functors

R Hom_{ℝ–HS_{r}}(ℝ(\underline{0}), −) \underrightarrow{∼} Γ̄_{ℝ}(−)

(3.2.3) Likewise, let V ∈ ℝ–HS_{r}^{+}. Then the real Frobenii τ_{j} act on Γ̃_{ℝ}(V). Let Γ̃_{ℝ}^{+}(V) be the invariants under the groups they generate. In the same way we obtain a functor Γ̄_{ℝ}^{+} on the derived category of ℝ–HS_{r}^{+} and an isomorphism

R Hom_{ℝ–HS_{r}^{+}}(ℝ(\underline{0}), −) \underrightarrow{∼} Γ̄_{ℝ}^{+}(−).

(3.2.4) Let us compute the Ext-groups for plectic Tate mixed Hodge structures. Now ℝ(\underline{n}) = ⊠_{j}ℝ(n_{j}) is the external tensor product of the Tate objects ℝ(n_{j}) in the category ℝ–HS of (usual) ℝ-Hodge structures. Therefore

R Hom_{ℝ–HS_{r}}(ℝ(\underline{0}), ℝ(\underline{n})) ≃ ⨂ R Hom_{ℝ–HS}(ℝ(0), ℝ(n_{j}))

Using the well-known formulae for the Ext-groups in ℝ–HS and ℝ–HS^{+}, we see at once that if some n_{j} is < 0, then R Hom_{ℝ–HS_{r}}(ℝ(\underline{0}), ℝ(\underline{n})) = 0. Otherwise it has exactly one non-vanishing cohomology group Ext_{ℝ–HS_{r}}^{d}(ℝ(\underline{0}), ℝ(\underline{n})) ≃ ℝ in degree d = #\{j | n_{j} > 0\}.

So passing to invariants under the τ_{j}, one sees that Ext_{ℝ–HS_{r}^{+}}^{d}(ℝ(\underline{0}), ℝ(\underline{n})) ≃ ℝ if each n_{j} is either zero or is odd and positive (with d as before), and vanishes otherwise.

3.3 Hilbert modular varieties

(3.3.1) Let F be a totally real field of degree r > 1, with ring of integers 𝔬, and Σ = Hom(F, ℝ) the set of real embeddings. Let X be a (complex, smooth, connected) Hilbert modular variety. Thus X = Γ\ℌ^{Σ} for a (sufficiently small) subgroup Γ ⊂ SL_{2}(𝔬) acting on r copies of ℌ, the complex upper half-place. In this section we will construct a mixed plectic ℝ-structure on H^{∗}(X, ℝ), compatible with the usual mixed Hodge structure, which is in some sense canonical (for example, one easily can show that it is preserved by Hecke operators).

(3.3.2) We first review the cohomology of X. For further details, see [7] for the topological and [6, Ch. III] for the Hodge theory. Let X ⟶^{j} \overline{X} be the minimal compactification of X. Its boundary is a finite set Y of points of X; let i: Y ⟶ X be the inclusion, and ν = #Y. One has the usual long exact sequence of boundary cohomology (with coefficients in a field E of characteristic zero, say):

⋯ → H_{c}^{n}(X, E) ⟶^{ρ^{n}} H^{n}(X, E) ⟶^{σ^{n}} H^{n}(Y, i^{∗}Rj_{∗}E) → …  (3.3.2.1)

Write H_{!}^{n}(X, E) = im(H_{c}^{n}(X, E) → H^{n}(X, E)) for the interior cohomology.

The boundary cohomology was computed by Harder [7, 8], together with the maps ρ^{n}, σ^{n}, using the Borel-Serre compactification. For each y ∈ Y, the Borel-Serre boundary component is a bundle over a real torus isogenous to 𝔬^{∗} ⊗_{ℤ} (ℝ/ℤ), with fibre isogenous to 𝔬 ⊗_{ℤ} (ℝ/ℤ). Its rational cohomology is

(R^{•}j_{∗}ℚ)_{y} ≃ ⋀_{ℤ}^{•} 𝔬^{∗} ⊗_{ℤ} (ℚe_{0} ⊕ ℚe_{r})

with e_{n} in degree n. Moreover the maps ρ^{n}, σ^{n} have the property:

• For r < n ≤ 2r − 1, ρ^{n} is injective.

• For r ≤ n < 2r − 1, σ^{n} is surjective.

• The image of σ^{2r−1} has codimension 1.

In particular, by Poincaré duality one has that H_{!}^{n}(X, ℚ) = H^{n}(X, ℚ) for 0 < n < r, and for r ≤ n ≤ 2r − 2 there is an exact sequence

0 → H_{!}^{n}(X, E) → H^{n}(X, E) → H^{n}(Y, i^{∗}Rj_{∗}E) → 0.

(3.3.3) Suppose from now on that E ⊂ ℝ. Then the exact sequence (3.3.2.1) is an exact sequence of mixed E-Hodge structure. The Hodge structure on the boundary cohomology is [6, §III.7]

(R^{n}j_{∗}E)_{y} = \begin{cases}⋀_{ℤ}^{n} 𝔬^{∗} ⊗_{ℤ} E(0) & (0 ≤ n ≤ r − 1) \\ ⋀_{ℤ}^{n−r} 𝔬^{∗} ⊗_{ℤ} E(−r) & (r ≤ n ≤ 2r − 1) \end{cases}

(3.3.4) The interior cohomology is pure, and is a direct sum of two parts. The first of these (denote H_{A}^{∗} in [7]) comes from algebraic classes associated to the r standard line bundles ℒ_{ι} (ι ∈ Σ) on X. Their cohomology classes ξ_{ι} belong to H_{!}^{2}(X, ℚ(1)). The E-subalgebra generated by {(2πi)^{−1}ξ_{ι}} is the sub-E-Hodge structure

H_{A}^{∗}(X, E) ≃ (E ⊕ E(−1))_{deg<2r}^{⊗Σ}.

of H_{!}^{∗}(X, E) (where E(−1) is in degree 2).

(3.3.5) The second part is the cuspidal cohomology H_{cusp}^{r}(X, E). It has dimension 2^{r} times the dimension of the space of holomorphic cusp forms of weight (2, …, 2) for Γ, and for E = ℝ can be interpreted as relative Lie algebra cohomology for 𝔤𝔩_{2}(ℝ)^{Σ}. As explained in [9, §2], there is a tensor product decomposition of the Lie algebra cohomology which puts on H_{cusp}^{r}(X, ℝ) a canonical pure plectic ℝ-structure, of plectic weight (1, …, 1).

(3.3.6) The real boundary cohomology can be given a canonical plectic Hodge structure, as a sum of copies of ℝ(0, …, 0) and ℝ(−1, …, −1).

(3.3.7) The algebraic part H_{A}^{∗} also carries a canonical plectic Hodge structure, for which ξ_{ι} (which is represented by a (1, 1)-form on the ι-th copy of ℌ) is of Hodge type (\underline{1}_{ι}, \underline{1}_{ι}). So H_{A}^{2m}(X, ℝ) is pure of simple weight 2m but has several plectic weights.

(3.3.8) The “Manin-Drinfeld principle” shows that H_{cusp}^{r}(X, ℚ) is (uniquely) a direct factor of H^{r}(X, ℚ) (not just of the interior cohomology), and thus the mixed Hodge structure on X is completely determined by the extension of the boundary cohomology by H_{A}^{∗}.

(3.3.9) In many degrees this is enough to put a canonical plectic Hodge structure on H^{n}(X, ℝ). The cases n = 0, 2r are trivial, and in degree 1 ≤ n < r, H^{n}(X, ℝ) = H_{!}^{n}(X, ℝ) so has a canonical plectic pure ℝ-Hodge structure. In odd degrees n with r < n < 2r − 1, H^{n}(X, ℝ) equals the boundary cohomology, so therefore has a canonical plectic Hodge structure. The same holds for n = 2r − 1 because of the exact sequence

0 = H_{!}^{2r−1}(X, ℝ) → H^{2r−1}(X, ℝ) → ⨁_{y∈Y} ℝ(−r) ⟶^{Σ} ℝ(−r) = H_{c}^{2r}(X, ℝ) → 0.

(3.3.10) To describe the mixed Hodge structure completely, it therefore suffices to consider the following possible extensions: if r is even

0 → H_{A}^{r}(X, ℝ) → H^{r}(X, ℝ)/H_{cusp}^{r}(X, ℝ) → ℝ(−r)^{ν} → 0  (3.3.10.1)

and for \frac{r}{2} < n < r

0 → H_{!}^{2n}(X, ℝ) = H_{A}^{2n}(X, ℝ) → H^{2n}(X, ℝ) → ⋀_{ℤ}^{2n−r} 𝔬^{∗} ⊗_{ℤ} ℝ(−r)^{ν} → 0. (3.3.10.2)

(3.3.11) Theorem. The extensions (3.3.10.1), (3.3.10.2) are (uniquely) split, with the exception of degree 2r − 2, when the class of the extension is described in (3.3.13) below.

For r = 2 this was proved by Caspar [3], using Eisenstein cohomology. The general case (which is proved the same way) will be found in the forthcoming PhD thesis of C. Davidescu. It follows that in these degrees, apart from 2r − 2, H^{n}(X, ℝ) carries a unique (unmixed) plectic ℝ-Hodge structure for which these sequences are exact sequences in ℝ–HS_{r}.

(3.3.12) In degree 2r − 2 the extension may be written (after fixing an orientation ∧^{r−1}0^{∗}/(torsion) ≃ ℤ) as

0 → ⨁_{ι: F →ℝ} ℝ(1) → H_{?}^{2r−2}(X, ℝ)(r) → Hom(𝔬^{∗}, ℤ) ⊗_{ℤ} ℝ(0) → 0

where H_{?}^{2r−2} = H^{2r−2} if r ≠ 2, and H_{?}^{2} = H^{2}/H_{cusp}^{2}. The ι-component of the direct sum is generated by ∧_{α≠ι}ξ_{α}. The extension is determined by the pushouts to these components, for each ι: F → ℝ:

0 → ℝ(1) → E_{ι} → Hom(𝔬^{∗}, ℤ) ⊗_{ℤ} ℝ(0) → 0.

The extension class of E_{ι} is an element of

Hom(𝔬^{∗}, ℤ) ⊗_{ℤ} Ext_{ℝ–HS}^{1}(ℝ(0), ℝ(1)) = Hom(𝔬^{∗}, ℝ).

(3.3.13) Theorem. The image of the class of E_{ι} in Hom(𝔬^{∗}, ℝ) is a non-zero rational multiple of the regulator homomorphism u ↦ log |ι(u)|.

(3.3.14) The subspace of cohomology generated by ∧_{α≠ι}ξ is isomorphic, as plectic Hodge structure, to the Tate object ℝ(−\underline{n}) with n_{α} = 1 (α ≠ ι), n_{ι} = 0. The boundary cohomology in degree (2r − 2) is a sum of copies of ℝ(−1, …, −1). To give a canonical plectic mixed ℝ-Hodge structure on H^{2r−2}(X, ℝ), it therefore is enough to show that each extension E_{ι} comes from a unique extension of plectic Hodge structures

0 → ℝ(\underline{1}_{ι}) → E_{ι} → Hom(𝔬^{∗}, ℤ) ⊗_{ℤ} ℝ(\underline{0}) → 0.

But the Künneth formula gives

Ext_{ℝ–HS_{r}}^{1}(ℝ(\underline{0}), ℝ(\underline{1}_{ι})) = Ext_{ℝ–HS}^{1}(ℝ(0), ℝ(1)) ⨂_{α≠ι} Hom_{ℝ–HS}(ℝ(0), ℝ(0)) ≃ ℝ

and so the forgetful map Ext_{ℝ–HS_{r}}^{1}(ℝ(\underline{0}), ℝ(\underline{1}_{ι})) → Ext_{ℝ–HS}^{1}(ℝ(0), ℝ(1)) is an isomorphism; so E_{ι} lifts to a unique element of ℝ–HS_{r}.

References

[1] A. A. Beilinson: Notes on absolute Hodge cohomology. In: Contemporary Mathematics vol. 55(1) (1986), 35–68

[2] A. Borel: Seminar on Transformation Groups. Annals of Mathematics Studies, No. 46. Princeton University Press, Princeton, N.J. 1960

[3] A. Caspar: Realisations of Kummer-Chern-Eisenstein-Motives. Manuscripta mathematica 122 (2007), 23–57

[4] P. Deligne: Théorie de Hodge II. Publ. Math. IHES, 40 (1971), 5–57

[5] — : Structures de Hodge mixtes réelles. In: Motives vol. I., Proc. Symp. Pure Math. vol. 55(1), American Math. Society, 1994, 509–514

[6] E. Freitag: Hilbert modular varieties. Springer, 1990

[7] G. Harder: On the cohomology of SL_{2}(𝔬). In: Lie groups and their representations. Adam Hilger Ltd., London, 1975, 139–150

[8] — : Eisenstein cohomology of arithmetic groups: the case GL_{2}. Inventiones Math. 89 (1987), 37–118

[9] J. Nekovář, A. J. Scholl: Introduction to plectic cohomology. In: Contemporary Mathematics vol. 664, American Math. Society, 2016, 321–337

[10] C. Weibel: Introduction to homological algebra. Cambridge University Press, 1994