Constructibility and duality for simple holonomic modules on complex symplectic manifolds

Masaki Kashiwara and Pierre Schapira

June 9, 2007

Abstract

Consider a complex symplectic manifold 𝔛 and the algebroid stack W_{𝔛} of deformation-quantization. For two regular holonomic W_{𝔛}- modules L_{i} (i = 0, 1) supported by smooth Lagrangian submanifolds, we prove that the complex RHom_{W_{𝔛}}(L_{1}, L_{0}) is perverse over the field W_{pt} and dual to the complex RHom_{W_{𝔛}}(L_{0}, L_{1}).

Introduction

Consider a complex symplectic manifold 𝔛. A local model for 𝔛 is an open subset of the cotangent bundle T^{∗}X to a complex manifold X, and T^{∗}X is endowed with the filtered sheaf of rings W_{T^{∗}X} of deformation-quantization. This sheaf of rings is similar to the sheaf E_{T^{∗}X} of microdifferential operators of Sato-Kawai-Kashiwara [18], but with an extra central parameter τ, a substitute to the lack of homogeneity (see § 1). This is an algebra over the field k = W_{pt}, a subfield of the field of formal Laurent series ℂ[[τ^{−1}, τ].

It would be tempting to glue the locally defined sheaves of algebras W_{T^{∗}X} to give rise to a sheaf on 𝔛, but the procedure fails, and one is lead to replace the notion of a sheaf of algebras by that of an “algebroid stack”. A canonical algebroid stack W_{𝔛} on 𝔛, locally equivalent to W_{T^{∗}X}, has been constructed by Polesello-Schapira [16] after Kontsevich [15] had treated the formal case (in the general setting of Poisson manifolds) by a different method.

In this paper, we study regular holonomic W_{𝔛}-modules supported by smooth Lagrangian submanifolds of 𝔛. For example, a regular holonomic module along the zero-section T_{X}^{∗}X of T^{∗}X is locally isomorphic to a finite sum of copies of the sheaf O_{X}^{τ} whose sections are series ∑_{−∞<j≤m} f_{j}τ^{j} (m ∈ ℤ), where the f_{j}’s are sections of O_{X} and satisfy certain growth conditions.

Mathematics Subject Classification: 46L65, 14A20, 32C38

Denote by D_{rh}^{b}(W_{𝔛}) the full subcategory of the bounded derived category of W_{𝔛}-modules on 𝔛 consisting of objects with regular holonomic cohomologies. The main theorem of this paper asserts that if L_{i} (i = 0, 1) are objects of D_{rh}^{b}(W_{𝔛}) supported by smooth Lagrangian manifolds Λ_{i} and if one sets F := RHom_{W_{𝔛}}(L_{1}, L_{0}), then F is ℂ-constructible over the field k, its microsupport is contained in the normal cone C(Λ_{0}, Λ_{1}) and F is dual over k to the object RHom_{W_{𝔛}}(L_{0}, L_{1}). If L_{0} and L_{1} are concentrated in degree 0, then F is perverse. We make the conjecture that the hypothesis on the smoothness of the Λ_{i}’s may be removed.

The strategy of our proof is as follows.

First, assuming only that L_{0} and L_{1} are coherent, we construct the canonical morphism

RHom_{W_{𝔛}}(L_{1}, L_{0}) ⟶ D_{𝔛}(RHom_{W_{𝔛}}(L_{0}, L_{1} [dim_{ℂ} 𝔛])), (0.1)

where D_{𝔛} is the duality functor for sheaves of k_{𝔛}-modules. By using a kind of Serre’s duality for the sheaf O_{X}^{τ}, we prove that (0.1) is an isomorphism as soon as RHom_{W_{𝔛}}(L_{0}, L_{1}) is constructible. For that purpose, we need to develop some functional analysis over nuclear algebras, in the line of Houzel [6].

In order to prove the constructibility result for F = RHom_{W_{𝔛}}(L_{1}, L_{0}), we may assume that 𝔛 = T^{∗}X, Λ_{0} = T_{X}^{∗}X, L_{0} = O_{X}^{τ} and Λ_{1} is the graph of the differential of a holomorphic function φ defined on X. Consider the sheaf of rings D_{X}[τ^{−1}] = D_{X} ⊗ ℂ[τ^{−1}]. We construct a coherent D_{X}[τ^{−1}]- module M which generates L_{1} and, setting F_{0} :=RHom_{D_{X}[τ^{−1}]}(M, O_{X}^{τ}(0)), we prove that the microsupport of F_{0} is a closed complex analytic Lagrangian subset of T^{∗}X contained in C(Λ_{0}, Λ_{1}). Using a deformation argument as in [7] and some functional analysis (extracted from [6]) over the base ring k_{0} = W_{pt}(0), we deduce that the fibers of the cohomology of F_{0} are finitely generated, and the result follows from the isomorphism F ≃ F_{0} ⊗_{k_{0}} k.

In this paper, we also consider a complex compact contact manifold 𝔜 and the algebroid stack E_{𝔜} of microdifferential operators on it. Denote by D_{rh}^{b}(E_{𝔜}) the ℂ-triangulated category consisting of E_{𝔜}-modules with regular holonomic cohomology. Using results of Kashiwara-Kawai [10], we prove that this category has finite Ext , admits a Serre functor in the sense of Bondal-Kapranov [2] and this functor is nothing but a shift by dim_{ℂ} 𝔜 + 1. In other words, D_{rh}^{b}(E_{𝔜}) is a Calabi-Yau category. Similar results hold over k for a complex compact symplectic manifold 𝔛.

1 W-modules on T^{∗}X

Let X be a complex manifold, π: T^{∗}X ⟶ X its cotangent bundle. The homogeneous symplectic manifold T^{∗}X is endowed with the ℂ^{×}–conic ℤfiltered sheaf of rings E_{T^{∗}X} of finite–order microdifferential operators, and its subring E_{T^{∗}X}(0) of operators of order ≤ 0 constructed in [18] (with other notations). We assume that the reader is familiar with the theory of Emodules, referring to [9] or [19] for an exposition.

On the symplectic manifold T^{∗}X there exists another (no more conic) sheaf of rings W_{T^{∗}X}, called ring of deformation-quantization by many authors (see Remark 1.2 below). The study of the relations between the ring E_{T^{∗}X} on a complex homogeneous symplectic manifold and the sheaf W_{T^{∗}X} on a complex symplectic manifold is systematically performed in [16], where it is shown in particular how quantized symplectic transformations act on W_{T^{∗}X}-modules. We follow here their presentation.

Let t ∈ ℂ be the coordinate and set

E_{T^{∗}(X×ℂ),t̂} = {P ∈ E_{T^{∗}(X×ℂ)} ; [P, ∂_{t}] = 0} .

Set T_{τ≠0}^{∗}(X × ℂ) = {(x, t; ξ, τ) ∈ T^{∗}(X × ℂ) ; τ ≠ 0}, and consider the map

ρ: T_{τ≠0}^{∗}(X × ℂ) ⟶ T^{∗}X (1.1)

given in local coordinates by ρ(x, t; ξ, τ) = (x; ξ/τ). The ring W_{T^{∗}X} on T^{∗}X is given by

W_{T^{∗}X} := ρ_{∗}(E_{X×ℂ,t̂}|_{T_{τ≠0}^{∗}(X×ℂ)}).

In a local symplectic coordinate system (x; u) on T^{∗}X, a section P of W_{T^{∗}X} on an open subset U is written as a formal series, called its total symbol:

σ_{tot}(P) = ∑_{−∞<j≤m} p_{j}(x; u)τ^{j}, m ∈ ℤ p_{j} ∈ O_{T^{∗}X}(U), (1.2)

with the condition

{for any compact subset K of U there exists a positive constant C_{K} such that sup_{K} |p_{j}| ≤ C_{K}^{−j}(−j)! for all j < 0.

(1.3)

The product is given by the Leibniz rule. If Q is an operator of total symbol σ_{tot}(Q), then

σ_{tot}(P ∘ Q) = ∑_{α∈ℕ^{n}} \frac{τ^{−|α|}}{α!} ∂_{u}^{α}σ_{tot}(P)∂_{x}^{α}σ_{tot}(Q).

Denote by W_{T^{∗}X}(m) the subsheaf of W_{T^{∗}X} consisting of sections P whose total symbol σ_{tot}(P) satisfies: p_{j} is zero for j > m. The ring W_{T^{∗}X} is ℤ-filtered by the W_{T^{∗}X}(m)’s, and W_{T^{∗}X}(0) is a subring. If P ∈ W_{T^{∗}X}(m) and P ∉ W_{T^{∗}X}(m − 1), P is said of order m. Hence 0 has order −∞. Then there is a well-defined principal symbol morphism, which does not depend on the local coordinate system on X:

σ_{m}: W_{T^{∗}X}(m) ⟶ O_{T^{∗}X} · τ^{m}. (1.4)

If P ∈ W_{T^{∗}X} has order m on a connected open subset of T^{∗}X, σ_{m}(P) is called the principal symbol of P and is denoted σ(P). Note that a section P in W_{T^{∗}X} is invertible on an open subset U of T^{∗}X if and only if its principal symbol is nowhere vanishing on U.

The principal symbol map induces an isomorphism of graded algebras

gr W_{T^{∗}X} \underrightarrow{∼} O_{T^{∗}X}[τ, τ^{−1}].

We set

k := W_{{pt}}, k(j) := W_{{pt}}(j) (j ∈ ℤ), k_{0} := k(0).

Hence, k is a field and an element a ∈ k is written as a formal series

a = ∑_{j≤m} a_{j}τ^{j}, a_{j} ∈ ℂ, m ∈ ℤ (1.5)

satisfying (1.3), that is,

{there exists a positive constant C such that |a_{j}| ≤ C^{−j}(−j)! for all j < 0.

(1.6)

Note that k_{0} is a discrete valuation ring.

Convention 1.1. In the sequel, we shall identify ∂_{t} and its total symbol τ. Hence, we consider τ as a section of W_{T^{∗}X}.

Note that

• W_{T^{∗}X} is flat over W_{T^{∗}X}(0) and in particular k is flat over k_{0},

• k_{0} is faithfully flat over ℂ[τ^{−1}],

• the sheaves of rings W_{T^{∗}X} and W_{T^{∗}X}(0) are right and left Noetherian (see [9, Appendix]) and in particular coherent, • if M is a coherent W_{T^{∗}X}-module, its support is a closed complex

analytic involutive (by Gabber’s theorem) subset of T^{∗}X,

• W_{T^{∗}X} is a k-algebra and W_{T^{∗}X}(0) is a k_{0}-algebra,

• there are natural monomorphisms of sheaves of ℂ-algebras

π^{−1}D_{X} ↪ E_{T^{∗}X} ↪ W_{T^{∗}X}. (1.7)

On an affine chart, morphism (1.7) is described on symbols as follows. To a section of E_{T^{∗}X} of total symbol ∑_{−∞<j≤m} p_{j}(x; ξ) (the p_{j}’s are homogeneous in ξ of degree j) one associates the section of W_{T^{∗}X} of total symbol ∑_{−∞<j≤m} p_{j}(x; u)τ^{j}, with u = τ^{−1}ξ.

Remark 1.2. (i) Many authors consider the filtered ring of formal operators, defined by

Ŵ_{T^{∗}X}(m) = \underleftarrow{lim}_{j≤m} W_{T^{∗}X}(m)/W_{T^{∗}X}(j) m ∈ ℤ, Ŵ_{T^{∗}X} = ⋃_{m} Ŵ_{T^{∗}X}(m).

Then

k̂ := Ŵ_{{pt}} ≃ ℂ[[τ^{−1}, τ], k̂_{0} := Ŵ_{{pt}}(0) ≃ ℂ[[τ^{−1}]].

Note that Ŵ_{T^{∗}X} is faithfully flat over W_{T^{∗}X} and Ŵ_{T^{∗}X} is flat over Ŵ_{T^{∗}X}(0). (ii) Many authors also prefer to use the symbol ℏ = τ^{−1} instead of τ.

Notation 1.3. Let I_{X} be the left ideal of W_{T^{∗}X} generated by the vector fields on X. We set

O_{X}^{τ} = W_{T^{∗}X}/I_{X}, O_{X}^{τ}(m) = W_{T^{∗}X}(m)/(I_{X} ∩ W_{T^{∗}X}(m)) (m ∈ ℤ).

Note that O_{X}^{τ} is a coherent W_{T^{∗}X}-module supported by the zero-section T_{X}^{∗}X. A section f(x, τ) of this module may be written as a series:

f(x, τ) = ∑_{−∞<j≤m} f_{j}(x)τ^{j}, m ∈ ℤ, (1.8)

the f_{j}’s satisfying the condition (1.3).

Also note that O_{X} is a direct summand of O_{X}^{τ} as a sheaf.

Lemma 1.4. After identifying X and X × {0} ⊂ X × ℂ, there is an isomorphism of sheaves of ℂ-vector spaces (not of algebras)

O_{X}^{τ}(0) ≃ O_{X×ℂ}|_{X×{0}}. (1.9)

Proof. By its construction, O_{X}^{τ}(0) is isomorphic to the sheaf of holomorphic microfunctions C_{X×{0}|X×ℂ}(0) of [18], and this last sheaf is isomorphic to O_{X×ℂ}|_{X×{0}} by loc. cit. q.e.d.

Note that isomorphism (1.9) corresponds to the map

O_{X}^{τ}(0) ∋ ∑_{j≤0} f_{j}τ^{j} ↦ ∑_{j≥0} f_{−j} \frac{t^{j}}{j!} ∈ O_{X×ℂ}|_{X×\{0\}}.

We shall use the following sheaves of rings. We set

(1.10) D_{X}[τ^{−1}] := D_{X} ⊗ ℂ[τ^{−1}], D_{X}[τ^{−1}, τ] := D_{X} ⊗ ℂ[τ, τ^{−1}].

Note that O_{X}^{τ}(0) is a left D_{X}[τ^{−1}]-module and O_{X}^{τ} a left D_{X}[τ^{−1}, τ]-module.

Lemma 1.5. (i) D_{X}[τ^{−1}] is a right and left Noetherian sheaf of ℂ-algebras,

(ii) if N ⊂ M are two coherent D_{X}[τ^{−1}]-modules and M_{0} a coherent D_{X}- submodule of M, then N ∩ M_{0} is D_{X}-coherent,

(iii) if M is a coherent D_{X}[τ^{−1}]-module and M_{0} a D_{X}-submodule of M of finite type, then M_{0} is D_{X}-coherent.

Proof. (i) follows from [9, Th. A. 3]. (ii) Set A := D_{X}[τ^{−1}] and let A_{n} ⊂ A be the D_{X}-submodule consisting of sections of order ≤ n with respect to τ^{−1}. We may reduce to the case where M = A^{N}. By [9, Th. A.29, Lem. A21], N ∩ A_{n} is D_{X}-coherent. Since {N ∩ A_{n} ∩ M_{0}}_{n} is an increasing sequence of coherent D_{X}-submodules of M_{0} and this last module is finitely generated, the sequence is stationary. (iii) Let us keep the notation in (ii). Here again, we may reduce to the case where M = A^{N}. Then M_{0} ∩ A_{n} is D_{X}-coherent, and the proof goes as in (ii). q.e.d.

Lemma 1.6. (i) D_{X}[τ^{−1}, τ] is flat over D_{X}[τ^{−1}],

(ii) if I is a finitely generated left ideal of D_{X}[τ^{−1}, τ], then I ∩ D_{X}[τ^{−1}] is a locally finitely generated left ideal of D_{X}[τ^{−1}].

Proof. (i) ℂ[τ, τ^{−1}] is flat over ℂ[τ^{−1}]. (ii) Let I_{0} ⊂ I be a coherent D_{X}[τ^{−1}]-module which generates I. Set J_{n} := τ^{n}I_{0} ∩ D_{X}[τ^{−1}]. Then I ∩ D_{X}[τ^{−1}] = ⋃_{n} J_{n} and this increasing sequence of coherent ideals of D_{X}[τ^{−1}] is locally stationary. q.e.d.

Lemma 1.7. W_{T^{∗}X} is flat over π^{−1}D_{X}[τ^{−1}, τ].

Proof. One proves that W_{T^{∗}X} is flat over π^{−1}D_{X}[τ^{−1}] exactly as one proves that the ring of microdifferential operators E_{T^{∗}X} is flat over π^{−1}D_{X}. Since we have D_{X}[τ^{−1}, τ] ⊗_{D_{X}[τ^{−1}]} M ≃ M for any D_{X}[τ^{−1}, τ]-module M, the result follows. q.e.d.

2 Regular holonomic W-modules

The following definition adapts a classical definition of [11] to W_{T^{∗}X}-modules (see also [4]).

Definition 2.1. Let Λ be a smooth Lagrangian submanifold of T^{∗}X.

(a) Let L(0) be a coherent W_{T^{∗}X}(0)-module supported by Λ. One says that L(0) is regular (resp. simple) along Λ if L(0)/L(−1) is a coherent O_{Λ}- module (resp. an invertible O_{Λ}-module). Here, L(−1) = W_{T^{∗}X}(−1)L(0).

(b) Let L be a coherent W_{T^{∗}X}-module supported by Λ. One says that L is regular (resp. simple) along Λ if there exists locally a coherent W_{T^{∗}X}(0)-submodule L(0) of L such that L(0) generates L over W_{T^{∗}X} and is regular (resp. simple) along Λ.

Note that in (b), L(0)/L(−1) is necessarily a locally free O_{Λ}-module.

Example 2.2. The sheaf O_{X}^{τ} is a simple W_{T^{∗}X}-module along the zerosection T_{X}^{∗}X.

The following result is easily proved (see [4]).

Proposition 2.3. Let Λ be a smooth Lagrangian submanifold of T^{∗}X and let M be a coherent W_{T^{∗}X}-module.

(i) If M is regular, then it is locally a finite direct sum of simple modules.

(ii) Any two W_{T^{∗}X}-modules simple along Λ are locally isomorphic. In particular, any simple module along T_{X}^{∗}X is locally isomorphic to O_{X}^{τ}.

(iii) If M, N are simple W_{T^{∗}X}-modules along Λ, then RHom_{W_{T^{∗}X}} (M, N) is concentrated in degree 0 and is a k-local system of rank one on Λ.

Definition 2.4. Let M be a coherent W_{T^{∗}X}-module and let Λ denote its support (a closed ℂ-analytic subset of T^{∗}X).

(i) We say that M is holonomic if Λ is Lagrangian.

(ii) Assume M is holonomic. We say that M is regular holonomic if there is an open subset U ⊂ 𝔛 such that U ∩ Λ is a dense subset of the regular locus Λ_{reg} of Λ and M|_{U} is regular along U ∩ Λ.

In other words, a holonomic module M is regular if it is so at the generic points of its support. Note that when Λ is smooth, Definition 2.4 is compatible with Definition 2.1. This follows from Gabber’s theorem. Indeed, we have the following theorem, analogous of [9, Th. 7.34]:

Theorem 2.5. Let U be an open subset of T^{∗}X, M a coherent W_{T^{∗}X}|_{U}- module and N ⊂ M a sub-W_{T^{∗}X}(0)-module. Assume that N is a small filtrant inductive limit of coherent W_{T^{∗}X}(0)-modules. Let V be the set of p ∈ U in a neighborhood of which N is W_{T^{∗}X}(0)-coherent. Then U \ V is an analytic involutive subset of U.

The fact that regularity of holonomic W-modules is a generic property follows as in loc. cit. Prop. 8.28.

3 W-modules on a complex symplectic manifold

We refer to [15] for the definition of an algebroid stack and to [3] for a more systematic study.

Let 𝕂 be a commutative unital ring and let X be a topological space. We denote by Mod(𝕂_{X}) the abelian category of sheaves of 𝕂-modules and by D^{b}(𝕂_{X}) its bounded derived category.

If A is a 𝕂-algebra, we denote by A^{+} the category with one object and having A as endomorphisms of this object. If A is a sheaf of 𝕂-algebras on X, we denote by A^{+} the 𝕂-linear stack associated with the prestack U ↦ A(U)^{+} (U open in X) and call it the 𝕂-algebroid stack associated with A. It is equivalent to the stack of right A-modules locally isomorphic to A, and A-linear homomorphisms.

The projective cotangent bundle P^{∗}Y to a complex manifold Y is endowed with the sheaf of rings E_{P^{∗}Y} of microdifferential operators. (This sheaf is the direct image of the sheaf E_{T^{∗}Y} of § 1 by the map T^{∗}Y \ T_{Y}^{∗} Y ⟶ P^{∗}Y.) A complex contact manifold 𝔜 is locally isomorphic to an open subset of a projective cotangent bundle P^{∗}Y and on such a contact manifold, a canonical algebroid stack E_{𝔜} locally equivalent to the stack associated with the sheaf of rings E_{P^{∗}Y} has been constructed in [8].

This construction has been adapted to the symplectic case by [16]. A complex symplectic manifold 𝔛 is locally isomorphic to the cotangent bundle T^{∗}X to a complex manifold X and a canonical algebroid stack W_{𝔛} locally equivalent to the stack associated with the sheaf of rings W_{T^{∗}X} of § 1 is constructed in loc. cit., after Kontsevich [15] had treated the general case of complex Poisson manifolds in the formal setting by a different approach.

Denote by 𝔛^{a} the complex manifold 𝔛 endowed with the symplectic form −ω, where ω is the symplectic form on 𝔛. There is a natural equivalence of algebroid stacks W_{𝔛^{a}} ≃ (W_{𝔛})^{op}.

Let 𝔛 and 𝔜 be two complex symplectic manifolds. There exist a natural k-algebroid stack W_{𝔛} ⊠ W_{𝔜} on 𝔛 × 𝔜 and a natural functor of k-algebroid stacks W_{𝔛} ⊠ W_{𝔜} ⟶ W_{𝔛×𝔜} which locally corresponds to the morphism of sheaves of rings W_{T^{∗}X} ⊠ W_{T^{∗}Y} ⟶ W_{T^{∗}(X×Y)}.

One sets

Mod(W_{𝔛}) = Fct_{k}(W_{𝔛}, 𝔐𝔬𝔡(k_{𝔛})),

where Fct_{k}(^{•} , ^{•}) denotes the k-linear category of k-linear functors of stacks and 𝔐𝔬𝔡(k_{𝔛}) is the stack of sheaves of k-modules on 𝔛. We denote by 𝔐𝔬𝔡(W_{𝔛}) the stack on 𝔛 given by U ↦ Mod(W_{𝔛}|_{U}).

Then Mod(W_{𝔛}) is a Grothendieck abelian category. We denote by D^{b}(W_{𝔛}) its bounded derived category and call an object of this derived category a W_{𝔛}-module on 𝔛, for short. Objects of Mod(W_{𝔛}) are described with some details in [4].

We denote by Hom_{W_{𝔛}} the hom-functor of the stack 𝔐𝔬𝔡(W_{𝔛}), a functor from Mod(W_{𝔛})^{op} × Mod(W_{𝔛}) to Mod(k_{𝔛}). The object RHom_{W_{𝔛}}(M, N) is thus well defined in D^{b}(k_{𝔛}) for two objects M and N of D^{b}(W_{𝔛}).

The natural functor W_{𝔛} ⊠ W_{𝔜} ⟶ W_{𝔛×𝔜} defines a functor of stacks

for : 𝔐𝔬𝔡(W_{𝔛×𝔜}) ⟶ 𝔐𝔬𝔡(W_{𝔛} ⊠ W_{𝔜}) (3.1)

and this last functor admits an adjoint (since it does locally). For M ∈ D^{b}(W_{𝔛}) and N ∈ D^{b}(W_{𝔜}), it thus exists a canonically defined object M\underline{⊠}N ∈ D^{b}(W_{𝔛×𝔜}) such that, locally, M\underline{⊠}N ≃ W_{T^{∗}(X×Y)} ⊠_{(W_{T^{∗}X}⊠W_{T^{∗}Y} )} (M ⊠ N) in D^{b}(W_{T^{∗}(X×Y)}).

Being local, the notions of coherent or holonomic, regular holonomic or simple object of Mod(W_{𝔛}) make sense. We denote by

• D_{coh}^{b}(W_{𝔛}) the full triangulated subcategory of D^{b}(W_{𝔛}) consisting of objects with coherent cohomologies,

• D_{hol}^{b}(W_{𝔛}) the full triangulated subcategory of D_{coh}^{b}(W_{𝔛}) consisting of objects with holonomic cohomologies, or equivalently, of objects with Lagrangian supports in 𝔛, • D_{rh}^{b}(W_{𝔛}) the full triangulated subcategory of D_{hol}^{b}(W_{𝔛}) consisting of

objects with regular holonomic cohomologies.

The support of an object M of D_{coh}^{b}(W_{𝔛}) is denoted by supp(M) and is also called its characteristic variety. This is a closed complex analytic involutive subset of 𝔛.

In the sequel, we shall denote by Δ_{𝔛} the diagonal of 𝔛 × 𝔛^{a} and identify it with 𝔛 by the first projection.

The next lemma follows from general considerations on stacks and its verification is left to the reader.

Lemma 3.1. There exists a canonical simple W_{𝔛×𝔛^{a}}-module C_{Δ_{𝔛}} supported by the diagonal Δ_{𝔛} such that if U is open in 𝔛 and isomorphic to an open subset V of a cotangent bundle T^{∗}X, then C_{Δ_{𝔛}}|_{U} is isomorphic to W_{T^{∗}X}|_{V} as a W_{T^{∗}X} ⊗ (W_{T^{∗}X})^{op}-module.

Definition 3.2. Let M ∈ D^{b}(W_{𝔛}). Its dual D′_{w}M ∈ D^{b}(W_{𝔛^{a}}) is given by

D′_{w}M:= RHom_{W_{𝔛}}(M, C_{Δ_{𝔛}}). (3.2)

Let Λ be a smooth Lagrangian submanifold of 𝔛. Consider the cohomology class c_{1/2} ∈ H^{2}(X; ℂ_{Λ}^{×}) defined as c_{1/2} = β(\frac{1}{2}α([Ω_{Λ}])) where [Ω_{Λ}] ∈ H^{1}(X; O_{Λ}^{×}) is the class of the line bundle Ω_{Λ} and α, β are the morphisms of the exact sequence

H^{1}(X; O_{Λ}^{×}) ⟶^{α} H^{1}(X; dO_{Λ}) ⟶^{β} H^{2}(X; ℂ_{Λ}^{×}).

The main theorem of [4] asserts that simple W_{𝔛}-modules along Λ are in one-to-one correspondence with twisted local systems of rank one on Λ with twist c_{1/2}.

Proposition 3.3. Let M, N be two objects of D_{coh}^{b}(W_{𝔛}). There is a natural isomorphism in D^{b}(k_{𝔛}):

RHom_{W_{𝔛}}(M, N) ≃ RHom_{W_{𝔛×𝔛^{a}}} (M\underline{⊠}D′_{w}N , C_{Δ_{𝔛}}).

Proof. We have the isomorphism in D^{b}(W_{𝔛}):

N ≃ RHom_{W_{𝔛^{a}}}(D′_{w}N , C_{Δ_{𝔛}}),

from which we deduce

RHom_{W_{𝔛×𝔛^{a}}} (M\underline{⊠}D′_{w}N , C_{Δ_{𝔛}}) ≃ RHom_{W_{𝔛}}(M, RHom_{W_{𝔛^{a}}}(D′_{w}N , C_{Δ_{𝔛}}))
≃ RHom_{W_{𝔛}}(M, N).

q.e.d.

4 Functional analysis I

In this section, we will use techniques elaborated by Houzel [6] and will follow his terminology. (See also Kiel-Verdier [14] for related results.)

We call a bornological convex ℂ-vector space (resp. ℂ-algebra), a bcspace (resp. bc-algebra) and we denote by Mod^{bc}(ℂ) the category of bc-spaces and bounded linear maps. This additive category admits small inductive and projective limits, but is not abelian.

Let A be a bc-algebra. We denote by Mod^{bc}(A) the additive category of bornological A-modules and bounded A-linear maps. For E, F ∈ Mod^{bc}(A), we set:

Bhom_{A}(E, F) = Hom_{Mod^{bc}(A)}(E, F),
E^{∨} = Bhom_{A}(E, A).

Let E ∈ Mod^{bc}(A) and let B ⊂ E be a convex circled bounded subset of E. For x ∈ E, one sets

||x||_{B} = inf_{x∈cB,c∈ℂ}|c|.

For u ∈ Bhom_{A}(E, F), B bounded in E and B′ convex circled bounded in F, one sets

||u||_{BB′} = sup_{x∈B} ||u(x)||_{B′}.

One says that a sequence {u_{n}}_{n} in Bhom_{A}(E, F) is bounded if for any bounded subset B ⊂ E there exists a convex circled bounded subset B′ ⊂ F

such that sup_{n} ||u_{n}||_{BB′} < ∞.

One says that u ∈ Bhom_{A}(E, F) is A-nuclear if there exist a bounded sequence {y_{n}}_{n} in F, a bounded sequence {u_{n}}_{n} in Bhom_{A}(E, A) and a summable sequence {λ_{n}}_{n} in ℝ_{≥0} such that

u(x) = ∑_{n} λ_{n} u_{n}(x) y_{n} for all x ∈ E.

For E, F ∈ Mod^{bc}(A), there is a natural structure of bc-space on E ⊗_{A} F and one denotes by E ˆ⊗_{A}F the completion of E ⊗_{A} F. Assuming F is complete, there is a natural linear map

E^{∨}ˆ⊗_{A}F ⟶ Bhom_{A}(E, F).

An element u ∈ Bhom_{A}(E, F) is A-nuclear if and only if it is in the image of E^{∨}ˆ⊗_{A}F.

Note that, if u: E ⟶ F is a nuclear ℂ-linear map, uˆ⊗ 1: E ˆ⊗ A ⟶ F ˆ⊗ A is A-nuclear.

Recall that a ℂ-vector space E is called a DFN-space if it is an inductive limit {(E_{n}, u_{n})}_{n∈ℕ} of Banach spaces such that the maps u_{n}: E_{n} ⟶ E_{n+1} are ℂ-nuclear and injective. Note that any bounded subset of E is contained in E_{n} for some n.

In the sequel, we will consider a DFN-algebra A and the full subcategory Mod^{dfn}(A) of Mod^{bc}(A) consisting of DFN-spaces. Note that any epimorphism u: E ⟶ F in Mod^{bc}(A) is semi-strict, that is, any bounded sequence in F is the image by u of a bounded sequence in E. Also note that for E and F in Mod^{dfn}(A), Bhom_{A}(E, F) is the subspace of Hom_{A}(E, F) consisting of continuous maps. Since the category Mod^{dfn}(A) is not abelian, we introduce the following definition, referring to [20] for a more systematic treatment of homological algebra in terms of quasi-abelian categories.

Definition 4.1. (i) Let A be a DFN-algebra. A complex 0 ⟶ E′ ⟶ E ⟶ E′′ ⟶ 0 in Mod^{dfn}(A) is called a short exact sequence if it is an exact sequence in Mod(A).

(ii) Let A and B be two DFN-algebra. An additive functor from Mod^{dfn}(A) to Mod^{dfn}(B) is called exact if it sends short exact sequences to short exact sequences.

The following result is well-known and follows from [5].

Proposition 4.2. Let A be a DFN-algebra. The functor ^{•} ˆ⊗ A: Mod^{dfn}(ℂ) ⟶ Mod^{dfn}(A) is exact.

Recall that a bc-algebra A is multiplicatively convex if for any bounded set B ⊂ A, there exist a constant c > 0 and a convex circled bounded set B′ such that B ⊂ c · B′ and B′ · B′ ⊂ B′.

As usual, for an additive category C, we denote by C^{b}(C) the category of bounded complexes in C and, for a ≤ b in ℤ, by C^{[a,b]}(C) the full subcategory consisting of complexes concentrated in degrees j ∈ [a, b]. We denote by K^{b}(C) the homotopy category associated with C^{b}(C). Finally, we denote by Ind(C) and Pro(C) the categories of ind-objects and pro-objects of C, respectively.

Theorem 4.3. Let A be a multiplicatively convex DFN-algebra and assume that A is a Noetherian ring (when forgetting the topology). Consider an inductive system {(E_{n}^{•}, u_{n}^{•})}_{n∈ℕ} in C^{[a,b]}(Mod^{dfn}(A)) for a ≤ b ∈ ℤ. Assume:

(i) u_{n}^{•}: E_{n}^{•} ⟶ E_{n+1}^{•} is a quasi-isomorphism for all n ≥ 0, (ii) u_{n}^{j}: E_{n}^{j} ⟶ E_{n+1}^{j} is A-nuclear for all j ∈ ℤ and all n ≥ 0. Then H^{j}(E_{n}^{•}) is finitely generated over A for all j ∈ ℤ and all n ≥ 0.

This is a particular case of [6, § 6 Th. 1, Prop. A.1]. Theorem 4.4. Let A be a DFN-algebra, and consider an inductive system {E_{n}^{•}, u_{n}^{•}}_{n∈ℕ} in C^{[a,b]}(Mod^{dfn}(A)) for a ≤ b in ℤ. Assume

(i) for each i ∈ ℤ and n ∈ ℕ, the map u_{n}^{i}: E_{n}^{i} ⟶ E_{n+1}^{i} is A-nuclear, (ii) for each i ∈ ℤ, “\underrightarrow{lim}”_{n} H^{i}(E_{n}^{•}) ≃ 0 in Ind(Mod(ℂ)).

Then “\underrightarrow{lim}”_{n} E_{n}^{•} ≃ 0 in Ind(K^{[a,b]}(Mod^{dfn}(A))).

First, we need a lemma. Lemma 4.5. Consider the solid diagram in Mod^{dfn}(A):

\xymatrix{E \ar[r]^{u} \ar@{.>}[d]_{v} & F \ar[d]^{v′} \\ E′ \ar[r]_{u′} & F′.}

Assume that u is A-nuclear and Im v′ ⊂ Im u′. Then there exists a morphism v: E ⟶ E′ making the whole diagram commutative. Proof. The morphism w: E′ ×_{F′} F ⟶ F is well defined in the category Mod^{dfn}(A) and is surjective by the hypothesis. We get a diagram

\xymatrix{ & E \ar[d]^{u} \\ E′ ×_{F′} F′ \ar@{->>}[r]_{w} & F,}

In this situation, the nuclear map u factors through E′×_{F′}F′ by [6, §4 Cor. 2]. q.e.d.

Proof of Theorem 4.4. We may assume that H^{i}(E_{n}^{•}) ⟶ H^{i}(E_{n+1}^{•}) is the zero morphism for all i ∈ ℤ and all n ∈ ℕ. Consider the solid diagram

\xymatrix{E_{n−1}^{b} \ar[r]^{u_{n−1}^{b}} \ar@{.>}[d]_{k_{n}^{b}} & E_{n}^{b} \ar[d]^{u_{n}^{b}} \\ E_{n+1}^{b−1} \ar[r]_{d_{n+1}^{b−1}} & F_{n+1}^{b}.}

Since H^{b}(E_{p}^{•}) ≃ Coker d_{p}^{b−1} for all p, Im u_{n}^{b} ⊂ Im d_{n+1}^{b−1}. Moreover, u_{n−1}^{b} is A-nuclear by the hypothesis. Therefore we may apply Lemma 4.5 and we obtain a map k_{n}^{b}: E_{n−1}^{b} ⟶ E_{n+1}^{b−1} making the whole diagram commutative. Set v_{n}^{i} = u_{n}^{i} ∘ u_{n−1}^{i} and h_{n}^{b} = d_{n+1}^{b−1} ∘ k_{n}^{b}. Consider the diagram

\xymatrix{⋯ \ar[r] & E_{n−1}^{b−2} \ar[r]^{d_{n−1}^{b−2}} \ar[d]^{v_{n}^{b−2}} & E_{n−1}^{b−1} \ar[r]^{d_{n−1}^{b−1}} \ar[d]^{v_{n}^{b−1}} & E_{n−1}^{b} \ar[r] \ar[d]^{v_{n}^{b}} \ar[dl]_{h_{n}^{b}} & 0 \\ ⋯ \ar[r] & E_{n+1}^{b−2} \ar[r]^{d_{n+1}^{b−2}} & E_{n+1}^{b−1} \ar[r]^{d_{n+1}^{b−1}} & E_{n+1}^{b} \ar[r] & 0.}

The morphisms v_{n}^{i}’s define a morphism of complex v_{n}: E_{n−1}^{•} ⟶ E_{n+1}^{•}. We define h_{n}^{i}: E_{n−1}^{i} ⟶ E_{n+1}^{i} by setting h_{n}^{i} = 0 for i ≠ b. Now denote by σ^{≤b−1}E_{n}^{•} the stupid truncated complex obtained by replacing E_{n}^{b} with 0. The morphism

v_{n} − h_{n} ∘ d_{n−1} − d_{n+1} ∘ h_{n}: E_{n−1}^{•} ⟶ E_{n+1}^{•}

factorizes through σ^{≤b−1}E_{n}^{•}. Hence, we get an isomorphism

“\underrightarrow{lim}”_{n} E_{n}^{•} \underrightarrow{∼} “\underrightarrow{lim}”_{n} σ^{≤b−1}E_{n}^{•}

in Ind(K^{[a,b]}(Mod^{dfn}(A))). By repeating this argument, we find the isomorphism “\underrightarrow{lim}”_{n} E_{n}^{•} ≃ “\underrightarrow{lim}”_{n} σ^{≤b−p}E_{n}^{•} for any p ∈ ℕ. This completes the proof.

q.e.d.

Theorem 4.6. Let A be a DFN-field, let a ≤ b in ℤ, consider an inductive system {E_{n}^{•}, u_{n}^{•}}_{n∈ℕ} in C^{[a,b]}(Mod^{dfn}(A)) and set F_{n}^{•} = (E_{n}^{•})^{∨} = Bhom_{A}(E_{n}^{•}, A). Assume

(i) for each i ∈ ℤ and n ∈ ℕ, the map u_{n}^{i}: E_{n}^{i} ⟶ E_{n+1}^{i} is A-nuclear, (ii) for each i ∈ ℤ, “\underrightarrow{lim}”_{n} H^{i}(E_{n}^{•}) belongs to Mod^{f}(A) (the category of finite-

dimensional A-vector spaces).

Then we have the isomorphism

(4.1)  “\underleftarrow{lim}”_{n} F_{n}^{•} ≃ “\underleftarrow{lim}”_{n} Hom_{A}(E_{n}^{•}, A) in Pro(K^{b}(Mod(A))).

In particular, for each i ∈ ℤ, “\underleftarrow{lim}”_{n} H^{−i}(F_{n}^{•}) belongs to Mod^{f}(A) and is dual

of “\underrightarrow{lim}”_{n} H^{i}(E_{n}^{•}).

Proof. Recall (see [13, Exe 15.1] that for an abelian category C and an inductive system {X_{j}}_{j∈J} in D^{[a,b]}(C) indexed by a small filtrant category J, if the object “\underrightarrow{lim}”_{j} H^{i}(X_{j}) of Ind(C) is representable for all i ∈ ℤ, then

“\underrightarrow{lim}”_{j} X_{j} ∈ Ind(D^{b}(C)) is representable.

Applying this result to our situation, we find that the object “\underrightarrow{lim}”_{n} E_{n}^{•}

of Ind(D^{b}(Mod(A))) is representable in D^{b}(Mod(A)).

Denote by L^{•} the complex given by L^{i} = “\underrightarrow{lim}”_{n} H^{i}(E_{n}^{•}) and zero differ-

entials. Since A is a field, there exists an isomorphism L^{•} \underrightarrow{∼} “\underrightarrow{lim}”_{n} E_{n}^{•} in

D^{b}(Mod(A)), hence a quasi-isomorphism u: L^{•} ⟶ “\underrightarrow{lim}”_{n} E_{n}^{•} in C^{b}(Mod(A)).

There exists n ∈ ℕ such that u factorizes through L^{•} ⟶ E_{n}^{•} for some n and we may assume n = 0. Since L^{•} belongs to C^{b}(Mod^{f}(A)), L^{•} ⟶ E_{0}^{•} is well defined in C^{b}(Mod^{dfn}(A)). For any n, let u_{n}: L^{•} ⟶ E_{n}^{•} be the induced morphism. Let G_{n}^{•} be the mapping cone of u_{n}. The morphism u_{n} is embedded in a distinguished triangle in K^{b}(Mod^{dfn}(A))

L^{•} ⟶^{u_{n}} E_{n}^{•} ⟶ G_{n}^{•} ⟶^{+1} .

By Theorem 4.4, “\underrightarrow{lim}”_{n} H^{i}(G_{n}^{•}) ≃ 0 implies “\underrightarrow{lim}”_{n} G_{n}^{•} ≃ 0 in Ind(K^{b}(Mod^{dfn}(A))).

Hence, for any K ∈ K^{b}(Mod^{dfn}(A)), the morphism

Hom_{K^{b}(Mod^{dfn}(A))}(K, L^{•}) ⟶ \underrightarrow{lim}_{n} Hom_{K^{b}(Mod^{dfn}(A))}(K, E_{n}^{•})

is an isomorphism. By the Yoneda lemma, we have thus obtained the isomorphism

L^{•} \underrightarrow{∼} “\underrightarrow{lim}”_{n} E_{n}^{•} in Ind(K^{b}(Mod^{dfn}(A))).

Since L^{•} belongs to K^{b}(Mod^{f}(A)), (L^{•})^{∨} ≃ Hom_{A}(L^{•}, A) and we obtain

“\underleftarrow{lim}”_{n} F_{n}^{•} ≃ (L^{•})^{∨} ≃ Hom_{A}(L^{•}, A) ≃ “\underleftarrow{lim}”_{n} Hom_{A}(E_{n}^{•}, A)

in Pro(K^{b}(Mod(A))). The isomorphisms for the cohomologies follow since “\underleftarrow{lim}” commutes with H^{−i}(^{•}) and Hom_{A}(^{•} , A). q.e.d.

A similar result to Theorem 4.6 holds for projective system.

Theorem 4.7. Let A be a DFN-field, let a ≤ b in ℤ, consider a projective system {F_{n}^{•}, v_{n}^{•}}_{n∈ℕ} in C^{[a,b]}(Mod^{dfn}(A)) and set E_{n}^{•} = Bhom_{A}(F_{n}^{•}, A). Assume

(i) for each i ∈ ℤ and n ∈ ℕ, the map v_{n}^{i}: F_{n+1}^{i} ⟶ F_{n}^{i} is A-nuclear,

(ii) for each i ∈ ℤ, “\underleftarrow{lim}”_{n} H^{i}(F_{n}^{•}) belongs to Mod^{f}(A) (the category of finite-

dimensional A-vector spaces).

Then we have the isomorphism

(4.2)  “\underrightarrow{lim}”_{n} E_{n}^{•} ≃ “\underrightarrow{lim}”_{n} Hom_{A}(F_{n}^{•}, A) in Ind(K^{b}(Mod(A))).

In particular, for each i ∈ ℤ, “\underrightarrow{lim}”_{n} H^{−i}(E_{n}^{•}) belongs to Mod^{f}(A) and is dual

of “\underleftarrow{lim}”_{n} H^{i}(F_{n}^{•}).

The proof being similar to the one of Theorem 4.6, we shall not repeat it.

In the course of § 5 below, we shall also need the next lemma.

Lemma 4.8. Let A be a DFN-algebra and let u: E_{0} ⟶ E_{1} be a ℂ-nuclear map of DFN-spaces. Recall that (^{•})^{∨} = Bhom_{A}(^{•} , A) and set (^{•})^{⋆} = Bhom_{ℂ}(^{•} , ℂ). Then the solid commutative diagram below may be completed with the dotted arrow as a commutative diagram:

\xymatrix{E_{1}^{⋆}ˆ⊗A \ar[r]^{u^{⋆} ˆ⊗A} \ar[d] & E_{0}^{⋆}ˆ⊗A \ar[d] \\ (E_{1}ˆ⊗A)^{∨} \ar[r]_{(uˆ⊗A)^{∨}} \ar@{.>}[ur] & (E_{0}ˆ⊗A)^{∨}.}

Proof. Consider the commutative diagram

\xymatrix{E_{0}^{⋆}ˆ⊗E_{1} \ar[d] \ar[r] & Bhom_{ℂ}(E_{0}, E_{1}) \ar[d] \\ Bhom_{A}((E_{1}ˆ⊗A)^{∨}, E_{0}^{⋆}ˆ⊗A) \ar[r] & Bhom_{A}((E_{1}ˆ⊗A)^{∨}, (E_{0}ˆ⊗A)^{∨}).}

Since u is nuclear, it is the image of an element of E_{0}^{⋆}ˆ⊗E_{1}. q.e.d.

5 Functional analysis II

This section will provide the framework for apply Theorems 4.3 and 4.6.

Here, X will denote a complex manifold. For a locally free O_{X}-module F of finite rank, we set

F^{τ} := F ⊗_{O_{X}} O_{X}^{τ} F^{τ} (0) := F ⊗_{O_{X}} O_{X}^{τ}(0).

Lemma 5.1. (i) The ℂ-algebra k_{0} is a multiplicatively convex DFN-algebra,

(ii) the ℂ-algebra k is a DFN-algebra,

(iii) the functor ^{•} ˆ⊗ k_{0} : Mod^{dfn}(ℂ) ⟶ Mod^{dfn}(k_{0}) is well-defined and exact,

(iv) the functor ^{•} ˆ⊗ k: Mod^{dfn}(ℂ) ⟶ Mod^{dfn}(k) is well-defined and exact, and is isomorphic to the functor (^{•} ˆ⊗ k_{0}) ⊗_{k_{0}} k.

Proof. (i) Define the subalgebra k_{0}(r) of k_{0} by

u = ∑_{j≤0} a_{j}τ^{j} belongs to k_{0}(r) if and only if

|u|_{r} := ∑_{j≤0} \frac{r^{−j}}{(−j)!} |a_{j}| < ∞. (5.1)

Then, for u,v in k_{0}(r), we have

|u · v|_{r} ≤ |u|_{r} · |v|_{r}.

Hence, (k_{0}(r), |^{•} |_{r}) is a Banach algebra and k_{0} is multiplicatively convex since it is the inductive limit of the k_{0}(r)’s. Moreover, k_{0} is a DFN-space because the linear maps k_{0}(r) ⟶ k_{0}(r′) are nuclear for 0 < r′ < r. (ii)–(iv) are clear. q.e.d.

Note that k is not multiplicatively convex.

Let M be a real analytic manifold, X a complexification of M. We denote as usual by A_{M} the sheaf on M of real analytic functions, that is, A_{M} = O_{X}|_{M}. Recall that, for K compact in M, Γ(K; A_{M}) is a DFN-space. We set

A_{M}^{τ} = O_{X}^{τ}|_{M}. (5.2)

Lemma 5.2. Let M be a real analytic manifold and K a compact subset of M. Then

(i) the sheaf A_{M}^{τ} is Γ(K; ^{•})-acyclic,

(ii) Γ(K; A_{M}^{τ}) ≃ Γ(K; A_{M})ˆ⊗ k,

(iii) the same result holds with A_{M}^{τ} and k replaced with A_{M}^{τ}(0) and k_{0}, respectively.

Proof. Applying Lemma 1.4, we have isomorphisms for each holomorphically convex compact subset K of X:

Γ(K; O_{X}^{τ}(0)) ≃ Γ(K × {0}; O_{X×ℂ})
≃ Γ(K; O_{X})ˆ⊗ O_{ℂ,0}
≃ Γ(K; O_{X})ˆ⊗ k_{0}.

This proves (i)–(ii) for A_{M}^{τ}(0) and k_{0}. The other case follows since A_{M}^{τ} ≃ A_{M}^{τ}(0) ⊗_{k_{0}} k. q.e.d.

Let us denote, as usual, by Db_{M} the sheaf of Schwartz’s distribution on M. Recall that Γ_{c}(M; Db_{M}) is a DFN-space.

Lemma 5.3. Let M be a real analytic manifold. There is a unique (up to unique isomorphism) sheaf of k-modules Db_{M}^{τ} on M which is soft and satisfies

Γ_{c}(U; Db_{M}^{τ}) ≃ Γ_{c}(U; Db_{M})ˆ⊗k

for each open subset U of M. The same result holds with k replaced with k_{0}. In this case, we denote by Db_{M}^{τ}(0) the sheaf of k_{0}-modules so obtained.

Proof. For two open subsets U_{0} and U_{1}, the sequence

0 ⟶ Γ_{c}(U_{0} ∩ U_{1}; Db_{M})ˆ⊗ k ⟶ (Γ_{c}(U_{0}; Db_{M})ˆ⊗ k) ⊕ (Γ_{c}(U_{1}; Db_{M})ˆ⊗ k) ⟶ Γ_{c}(U_{0} ∪ U_{1}; Db_{M})ˆ⊗ k ⟶ 0

is exact, and similarly with k replaced with k_{0}. The results then easily follow. q.e.d.

Denote by \overline{X} the complex conjugate manifold to X and by X_{ℝ} the real underlying manifold, identified with the diagonal of X × \overline{X}. We shall write for short A_{X}^{τ} and Db_{X}^{τ} instead of A_{X_{ℝ}}^{τ} and Db_{X_{ℝ}}^{τ}, respectively. We set A_{X}^{(p,q),τ} = A_{X}^{(p,q)} ⊗_{A_{X}} A_{X}^{τ}, and similarly with Db_{X} instead of A_{X}.

Consider the Dolbeault-Grothendieck complexes of sheaves of k-modules

0 ⟶ A_{X}^{(0,0),τ} ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} A_{X}^{(0,d),τ} ⟶ 0, (5.3)

0 ⟶ Db_{X}^{(0,0),τ} ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} Db_{X}^{(0,d),τ} ⟶ 0. (5.4)

Lemma 5.4. Both complexes (5.3) and (5.4) are qis to O_{X}^{τ}. The same result holds when replacing A_{X}^{τ}, Db_{X}^{τ} and O_{X}^{τ} with A_{X}^{τ}(0), Db_{X}^{τ}(0) and O_{X}^{τ}(0), respectively.

The easy proof is left to the reader.

Lemma 5.5. Let X be a complex Stein manifold, K a holomorphically convex compact subset and F a locally free O_{X}-module of finite rank. Then

(i) one has an isomorphism Γ(K; F^{τ}) ≃ Γ(K; F)ˆ⊗ k,

(ii) the ℂ-vector space Γ(K; F^{τ}) is naturally endowed with a topology of DFN-space,

(iii) RΓ(K; F^{τ}) is concentrated in degree 0,

(iv) for K_{0} and K_{1} two compact subsets of X such that K_{0} is contained in the interior of K_{1}, the morphism Γ(K_{1}; F^{τ}) ⟶ Γ(K_{0}; F^{τ}) is k-nuclear,

(v) the same results hold with F^{τ} and k replaced with F^{τ} (0) and k_{0}, respectively.

Proof. (i)–(iii) By the hypothesis, the sequence

0 ⟶ Γ(K; O_{X}) ⟶ Γ(K; A_{X}^{(0,0)}) ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} Γ(K; A_{X}^{(0,d)}) ⟶ 0 (5.5)

is exact. It remains exact after applying the functor ^{•} ˆ⊗ k. The result will follow when comparing the sequence so obtained with the complex

0 ⟶ Γ(K; O_{X}^{τ}) ⟶ Γ(K; A_{X}^{(0,0),τ}) ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} Γ(K; A_{X}^{(0,d),τ}) ⟶ 0, (5.6)

The case of F is treated similarly, replacing A_{X} with F ⊗_{O_{X}} A_{X}. (iv) follows from (i) and the corresponding result for O_{X}. (iv) The proof for F^{τ} (0) and k_{0} is similar. q.e.d.

Lemma 5.6. Let X be a complex manifold of complex dimension d and let F be a locally free O_{X}-module of finite rank. Assume that X is Stein. Then

(i) one has the isomorphism H_{c}^{d}(X; F^{τ}) ≃ H_{c}^{d}(X; F)ˆ⊗k,

(ii) the ℂ-vector space H_{c}^{d}(X; F^{τ}) is naturally endowed with a topology of DFN-space,

(iii) RΓ_{c}(X; F_{X}^{τ}) is concentrated in degree d,

(iv) for U_{0} and U_{1} two Stein open subset of X with U_{0} ⊂⊂ U_{1}, the map H_{c}^{d}(U_{0}; F_{X}^{τ}) ⟶ H_{c}^{d}(U_{1}; F_{X}^{τ}) is k-nuclear,

(v) the same results hold with F^{τ} and k replaced with F^{τ} (0) and k_{0}, respectively.

Proof. The proof is similar to that of Lemma 5.5. By the hypothesis, the sequence

0 ⟶ Γ_{c}(X; Db_{X}^{(0,0)}) ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} Γ_{c}(X; Db_{X}^{(0,d)}) ⟶ H_{c}^{d}(X; O_{X}) ⟶ 0 (5.7)

is exact and H_{c}^{d}(X; O_{X}) is a DFN-space. This sequence will remain exact after applying the functor ^{•} ˆ⊗k. The result will follow when comparing the obtained sequence with

0 ⟶ Γ_{c}(X; Db_{X}^{(0,0),τ}) ⟶^{\overline{∂}} ⋯ ⟶^{\overline{∂}} Γ_{c}(X; Db_{X}^{(0,d),τ}) ⟶ H_{c}^{d}(X; O_{X}^{τ}) ⟶ 0. (5.8)

The case of F is treated similarly, replacing Db_{X} with F ⊗_{O_{X}} Db_{X}. q.e.d.

Proposition 5.7. Let f: X ⟶ Y be a morphism of complex manifolds of complex dimension d_{X} and d_{Y}, respectively. There is a k-linear morphism

∫_{f}: Rf_{!}Ω_{X}^{τ} [d_{X}] ⟶ Ω_{Y}^{τ} [d_{Y}] (5.9)

functorial with respect to f, which sends Rf_{!}Ω_{X}^{τ}(0) [d_{X}] to Ω_{Y}^{τ} (0) [d_{Y}] and which induces the classical integration morphism ∫_{f}: Rf_{!}Ω_{X} [d_{X}] ⟶ Ω_{Y} [d_{Y}] when identifying Ω_{X} and Ω_{Y} with a direct summand of Ω_{X}^{τ} and Ω_{Y}^{τ}, respectively.

In particular, there is a k-linear morphism

∫_{X}: H_{c}^{d_{X}} (X; Ω_{X}^{τ}) ⟶ k (5.10)

which induces the classical residues morphism H_{c}^{d_{X}} (X; Ω_{X}) ⟶ ℂ when identifying H_{c}^{d_{X}} (X; Ω_{X}) (resp. ℂ) with a direct summand of H_{c}^{d_{X}} (X; Ω_{X}^{τ}) (resp. k).

In the particular case where X is Stein and Y = pt, this follows easily from Lemma 5.6. Since we shall not use the general case, we leave the proof to the reader.

Proposition 5.8. Let X be a Stein complex manifold of complex dimension d and let K be a holomorphically convex compact subset of X. Then the pairing for a Stein open subset U of X

(5.11)  H_{c}^{d}(U; Ω_{X}^{τ}) × Γ(U; O_{X}^{τ}) ⟶ k, (fdx, g) ↦ ∫_{U} gfdx,

defines the isomorphisms “\underleftarrow{lim}”_{U⊃K} Bhom_{k}(Γ(\overline{U}; O_{X}^{τ}), k) ≃ “\underleftarrow{lim}”_{U⊃K} H_{c}^{d}(U; Ω_{X}^{τ})

and “\underrightarrow{lim}”_{U⊃K} Bhom_{k}(H_{c}^{d}(U; Ω_{X}^{τ}), k) ≃ “\underrightarrow{lim}”_{U⊃K} Γ(\overline{U}; O_{X}^{τ}). Here, U ranges over

the family of Stein open neighborhoods of K.

Proof. We shall prove the first isomorphism, the other case being similar. By Lemmas 5.5, 5.6 and 4.8, we have:

“\underleftarrow{lim}”_{U⊃K} Bhom_{k}(Γ(\overline{U}; O_{X}^{τ}), k) ≃ “\underleftarrow{lim}”_{U⊃K}(O_{X}(\overline{U})ˆ⊗ k)^{∨}
≃ “\underleftarrow{lim}”_{U⊃K}(O_{X}(\overline{U}))^{⋆}ˆ⊗ k
≃ “\underleftarrow{lim}”_{U⊃K}(O_{X}(U))^{⋆}ˆ⊗ k
≃ “\underleftarrow{lim}”_{U⊃K}(H_{c}^{d}(U; Ω_{X}))ˆ⊗ k
≃ “\underleftarrow{lim}”_{U⊃K} H_{c}^{d}(U; Ω_{X}^{τ}).

q.e.d.

In the proof of Corollary 7.6, we shall need the following result.

Lemma 5.9. Let Y be a smooth closed submanifold of codimension l of X. Then H^{j}(RΓ_{Y} (O_{X}^{τ})) and H^{j}(RΓ_{Y} (O_{X}^{τ}(0))) vanish for j < l.

Proof. Since the problem is local, we may assume that X = Y × ℂ^{l} where Y ⊂ X is identified with Y × {0}.

Let U_{n} be the open ball of ℂ^{l} centered at 0 with radius 1/n. Using the Mittag-Leffler theorem (see [12, Prop. 2.7.1]), it is enough to prove that for any holomorphically convex compact subset K of Y

H_{c}^{j}(K × U_{n}; O_{X}^{τ}) = 0 for j < l,

and similarly with O_{X}^{τ}(0). It is enough to prove the result for the sheaf O_{X}^{τ}(0). Then we may replace O_{X}^{τ}(0) with the sheaf O_{X×ℂ}|_{X×{0}} and we are reduced to the well-known result

H_{c}^{j}(K × U_{n} × {0}; O_{X×ℂ}) = 0 for j < l.

q.e.d.

Remark 5.10. Related results have been obtained, in a slightly different framework, in [17]

6 Duality for W-modules

We mainly follow the notations of [12]. Let X be a real manifold and 𝕂 a field. For F ∈ D^{b}(𝕂_{X}), we denote by SS(F) its microsupport, a closed ℝ^{+}-conic subset of T^{∗}X. Recall that this set is involutive (see loc. cit. Def. 6.5.1).

We denote by D_{X} the duality functor:

D_{X}: (D^{b}(𝕂_{X}))^{op} ⟶ D^{b}(𝕂_{X}), F ↦ RHom_{𝕂_{X}}(F, ω_{X}),

where ω_{X} is the dualizing complex.

Assume now that X is a complex manifold. We denote by dim_{ℂ} X its complex dimension. We identify the orientation sheaf on X with the constant sheaf, and the dualizing complex ω_{X} with 𝕂_{X} [2 dim_{ℂ} X].

Recall that an object F ∈ D^{b}(𝕂_{X}) is weakly-ℂ-constructible if there exists a complex analytic stratification X = ⨆_{α∈A} X_{α} such that H^{j}(F)|_{X_{α}} is locally constant for all j ∈ ℤ and all α ∈ A. The object F is ℂ-constructible if moreover H^{j}(F)_{x} is finite-dimensional for all x ∈ X and all j ∈ ℤ. We denote by D_{w−ℂ−c}^{b}(𝕂_{X}) the full subcategory of D^{b}(𝕂_{X}) consisting of weakly-ℂ-constructible objects and by D_{ℂ−c}^{b}(𝕂_{X}) the full subcategory consisting of ℂ-constructible objects.

Recall ([12]) that F ∈ D^{b}(𝕂_{X}) is weakly-ℂ-constructible if and only if its microsupport is a closed ℂ-conic complex analytic Lagrangian subset of T^{∗}X or, equivalently, if it is contained in a closed ℂ-conic complex analytic isotropic subset of T^{∗}X.

From now on, our base field is k.

Theorem 6.1. Let 𝔛 be a complex symplectic manifold and let L_{0} and L_{1} be two objects of D_{coh}^{b}(W_{𝔛}).

(i) There is a natural morphism

RHom_{W_{𝔛}}(L_{1}, L_{0}) ⟶ D_{𝔛}(RHom_{W_{𝔛}}(L_{0}, L_{1} [dim_{ℂ} 𝔛])). (6.1)

(ii) Assume that RHom_{W_{𝔛}}(L_{0}, L_{1}) belongs to D_{ℂ−c}^{b}(k_{𝔛}). Then the morphism (6.1) is an isomorphism. In particular, RHom_{W_{𝔛}}(L_{1}, L_{0}) belongs to D_{ℂ−c}^{b}(k_{𝔛}).

Proof. By Proposition 3.3, we may assume that L_{0} is a simple module along a smooth Lagrangian manifold Λ_{0}. In this case, k_{Λ_{0}} ⟶ RHom_{W_{𝔛}}(L_{0}, L_{0}) is an isomorphism.

(i) The natural k-linear morphism

RHom_{W_{𝔛}}(L_{1}, L_{0}) ⊗_{k} RHom_{W_{𝔛}}(L_{0}, L_{1}) ⟶ RHom_{W_{𝔛}}(L_{0}, L_{0}) ≃ k_{Λ_{0}}

defines the morphism

RHom_{W_{𝔛}}(L_{0}, L_{1}) ⟶ RHom_{k_{Λ_{0}}} (RHom_{W_{𝔛}}(L_{1}, L_{0}), k_{Λ_{0}}). (6.2)

To conclude, remark that for an object F ∈ D^{b}(k_{𝔛}) supported by Λ_{0}, we have

D_{𝔛}F ≃ RHom_{k_{Λ_{0}}} (F, k_{Λ_{0}}) [dim_{ℂ} 𝔛].

(ii) Let us prove that (6.2) is an isomorphism by adapting the proof of [7, Prop. 5.1].

Since this is a local problem, we may assume that 𝔛 = T^{∗}X, X is an open subset of ℂ^{d}, Λ_{0} = T_{X}^{∗}X and L_{0} = O_{X}^{τ}. Since RHom_{W_{𝔛}}(O_{X}^{τ}, L_{1}) is constructible, we are reduced to prove the isomorphisms for each x ∈ X

H^{j}(RHom_{W_{𝔛}}(L_{1}, O_{X}^{τ}))_{x} ≃ \underrightarrow{lim}_{U∋x}(H^{j}RΓ_{c}(U; RHom_{W_{𝔛}}(O_{X}^{τ}, L_{1})) [2d])^{⋆} (6.3)

where U ranges over the family of Stein open neighborhoods of x and ^{⋆} denotes the duality functor in the category of k-vector spaces.

We choose a finite free resolution of L_{1} on a neighborhood of x:

0 ⟶ W_{T^{∗}X}^{N_{r}} ⟶^{·P_{r−1}} ⋯ ⟶^{·P_{0}} W_{T^{∗}X}^{N_{0}} ⟶ L_{1} ⟶ 0 for some r ≥ 0.

For a sufficiently small holomorphically convex compact neighborhood K of x, the object RΓ(K; RHom_{W_{T^{∗}X}} (L_{1}, O_{X}^{τ})) is represented by the complex

E^{•}(K):= 0 ⟶ (Γ(K; O_{X}^{τ}))^{N_{0}} ⟶^{P_{0}·} ⋯ ⟶^{P_{r−1}·} (Γ(K; O_{X}^{τ}))^{N_{r}} ⟶ 0,

where (Γ(K; O_{X}^{τ}))^{N_{0}} stands in degree 0. Since

RHom_{W_{T^{∗}X}} (O_{X}^{τ}, W_{T^{∗}X} [d]) ≃ Ω_{X}^{τ},

the object RHom_{W}(O_{X}^{τ}, L_{1} [d]) is represented by the complex

0 ⟶ (Ω_{X}^{τ})^{N_{r}} ⟶^{·P_{r−1}} ⋯ ⟶^{·P_{0}} (Ω_{X}^{τ})^{N_{0}} ⟶ 0,

where (Ω_{X}^{τ})^{N_{0}} stands in degree 0. Hence, for a sufficiently small Stein open neighborhood U of x, RΓ_{c}(U; RHom_{W}(O_{X}^{τ}, L_{1} [2d])) is represented by the complex

F_{c}^{•}(U):= 0 ⟶ (H_{c}^{d}(U; Ω_{X}^{τ}))^{N_{r}} ⟶^{·P_{r−1}} ⋯ ⟶^{·P_{0}} (H_{c}^{d}(U; Ω_{X}^{τ}))^{N_{0}} ⟶ 0

where (H_{c}^{d}(U; Ω_{X}^{τ}))^{N_{0}} stands in degree 0.

Let U_{n} be the open ball of X centered at x with radius 1/n. By the hypothesis, all morphisms

F_{c}^{•}(U_{p}) ⟶ F_{c}^{•}(U_{n})

are quasi-isomorphisms for p ≥ n ≫ 0, and the cohomologies are finitedimensional over k. Therefore, the hypotheses of Theorem 4.7 are satisfied by Lemma 5.6 and we get

“\underrightarrow{lim}”_{n} Hom_{k}(F_{c}^{•}(U_{n}), k) ≃ “\underrightarrow{lim}”_{n}(F_{c}^{•}(U_{n}))^{∨} in Ind(K^{b}(Mod^{dfn}(k))).

Applying Proposition 5.8, we obtain

H^{j}(RHom_{W}(L_{1}, O_{X}^{τ}))_{x} ≃ \underrightarrow{lim}_{n} H^{j}(E^{•}(\overline{U}_{n}))
≃ \underrightarrow{lim}_{n} H^{j}Hom_{k}(F_{c}^{•}(U_{n}), k)
≃ \underrightarrow{lim}_{n}(H^{j}RΓ_{c}(U; RHom_{W_{𝔛}}(O_{X}^{τ}, L_{1})) [2d])^{⋆}.

q.e.d.

Corollary 6.2. In the situation of Theorem 6.1 (ii), assume moreover that 𝔛 is compact of complex dimension 2n. Then the k-vector spaces Ext_{W_{𝔛}}^{j}(L_{1}, L_{0}) and Ext_{W_{𝔛}}^{2n−j}(L_{0}, L_{1}) are finite-dimensional and dual to each other.

7 Statement of the main theorem

For two subsets V and S of the real manifold X, the normal cone C(S, V) is well defined in T X. If V is a smooth and closed submanifold of X, one denotes by C_{V} (S) the image of C(S, V) in the normal bundle T_{V} X.

Consider now a complex symplectic manifold (𝔛, ω). The 2-form ω gives the Hamiltonian isomorphism H from the cotangent to the tangent bundle to 𝔛:

H: T^{∗}𝔛 \underrightarrow{∼} T 𝔛, ⟨θ, v⟩ = ω(v, H(θ)), v ∈ T 𝔛, θ ∈ T^{∗}𝔛. (7.1)

For a smooth Lagrangian submanifold Λ of 𝔛, the isomorphism (7.1) induces an isomorphism between the normal bundle to Λ in 𝔛 and its cotangent bundle:

T_{Λ}𝔛 ≃ T^{∗}Λ. (7.2)

Let us recall a few notations and conventions (see [12]). For a complex manifold X and a complex analytic subvariety Z of X, one denotes by Z_{reg} the smooth locus of Z, a complex submanifold of X. For a holomorphic p-form θ on X, one says that θ vanishes on Z and one writes θ|_{Z} = 0 if θ|_{Z_{reg}} = 0.

Proposition 7.1. Let 𝔛 be a complex symplectic manifold and let Λ_{0} and Λ_{1} be two closed complex analytic isotropic subvarieties of 𝔛. Then, after identifying T 𝔛 and T^{∗}𝔛 by (7.1), the normal cone C(Λ_{0}, Λ_{1}) is a complex analytic ℂ^{×}-conic isotropic subvariety of T^{∗}𝔛.

Note that the same result holds for real analytic symplectic manifolds, replacing “complex analytic variety” with “subanalytic subset” and “ℂ^{×}- conic” with “ℝ^{+}-conic”.

First we need two lemmas.

Lemma 7.2. Let X be a complex manifold and θ a p-form on X. Let Z ⊂ Y be closed subvarieties of X. If θ|_{Y} = 0, then θ|_{Z} = 0.

Proof. By Whitney’s theorem, we can find an open dense subset Z′ of Z_{reg} such that

{for any sequence {y_{n}}_{n} in Y_{reg} such that it converges to a point z ∈ Z′ and {T_{y_{n}}Y }_{n} converges to a linear subspace τ ⊂ T_{z}X, τ contains T_{z}Z′.

Since θ vanishes on T_{y_{n}}Y, it vanishes also on τ and hence on T_{z}Z′. q.e.d.

Lemma 7.3. Let X be a complex manifold, Y a closed complex subvariety of X and f: X ⟶ ℂ a holomorphic function. Set Z:=f^{−1}(0), Y′:=\overline{(Y \ Z)}∩Z. Consider a p-form η, a (p − 1)-form θ on X and set

ω = df ∧ θ + fη.

Assume that ω|_{Y} = 0. Then θ|_{Y′} = 0 and η|_{Y′} = 0.

Proof. We may assume that Y = \overline{(Y \ Z)}.

Using Hironaka’s desingularization theorem, we may find a smooth manifold Ỹ and a proper morphism p: Ỹ ⟶ Y such that, in a neighborhood of each point of Ỹ, p^{∗}f may be written in a local coordinate system (y_{1}, …, y_{n}) as a product ∏_{i=1}^{n} y_{i}^{a_{i}}, where the a_{i}’s are non-negative integers. Let Z̃ = p^{−1}(Z). Then \overline{(Ỹ \ Z̃)} ∩ Z̃ ⟶ Y′ is proper and surjective.

Hence, we may assume from the beginning that Y is smooth and then Y = X. Moreover, it is enough to prove the result at generic points of Y′. Hence, we may assume, setting (y_{1}, …, y_{n}) = (t, x) (x = (y_{2}, …, y_{n})), that f(t, x) = t^{a} for some a > 0. Write

θ = tθ_{0} + dt ∧ θ_{1} + θ_{2},

where θ_{1} and θ_{2} depend only on x and dx.

By the hypothesis,

0 = df ∧ θ + fη = at^{a}dt ∧ θ_{0} + at^{a−1}dt ∧ θ_{2} + t^{a}η.

It follows that θ_{2} is identically zero. Hence:

θ = tθ_{0} + dt ∧ θ_{1}, η = −adt ∧ θ_{0}.

Therefore, θ|_{t=0} = η|_{t=0} = 0. q.e.d.

Proof of Proposition 7.1. Recall that 𝔛^{a} denotes the complex manifold 𝔛 endowed with the symplectic form −ω and that Δ denotes the diagonal of 𝔛 × 𝔛^{a}. Using the isomorphisms

T 𝔛 ≃ T_{Δ}(𝔛 × 𝔛^{a}),
C(Λ_{0}, Λ_{1}) ≃ C(Δ, Λ_{0} × Λ_{1}^{a}),

(the first isomorphism is associated with the first projection on 𝔛 × 𝔛^{a}) we are reduced to prove the result when Λ_{0} is smooth.

Let (x, u) be a local symplectic coordinate system on X such that

Λ_{0} = {(x; u) ; u = 0} , ω = ∑_{i=1}^{n} du_{i} ∧ dx_{i}.

Consider the normal deformation 𝔛̃_{Λ_{0}} of 𝔛 along Λ_{0}. Recall that we have a diagram

\xymatrix{𝔛 & 𝔛̃_{Λ_{0}} \ar[l]_{p} \ar[r]^{t} & ℂ \\ Λ_{1} \ar@{^{(}->}[u] & p^{−1}Λ_{1} \ar@{^{(}->}[u] \ar[l]_{p}}

such that, denoting by (x; ξ, t) the coordinates on 𝔛̃_{Λ_{0}},

p(x; ξ, t) = (x; tξ),
T_{Λ_{0}}𝔛 ≃ {(x; ξ, t) ∈ 𝔛̃_{Λ_{0}} ; t = 0} ,
C_{Λ_{0}}(Λ_{1}) ≃ \overline{p^{−1}(Λ_{1}) \ t^{−1}(0)} ∩ t^{−1}(0).

Clearly, C_{Λ_{0}}(Λ_{1}) is a complex analytic variety. Moreover,

p^{∗}ω = ∑_{i=1}^{n} d(tξ_{i}) ∧ dx_{i} = dt ∧ (∑_{i=1}^{n} ξ_{i} ∧ dx_{i}) + t ∑_{i=1}^{n} dξ_{i} ∧ dx_{i}.

Since p^{∗}ω vanishes on (p^{−1}Λ_{1})_{reg}, ∑_{i=1}^{n} dξ_{i} ∧ dx_{i} vanishes on C_{Λ_{0}}(Λ_{1}) by Lemma 7.3. q.e.d.

Theorem 7.4. Let 𝔛 be a complex symplectic manifold and let L_{i} (i = 0, 1) be two objects of D_{rh}^{b}(W_{𝔛}) supported by smooth Lagrangian submanifolds Λ_{i}. Then

(i) the object RHom_{W_{𝔛}}(L_{1}, L_{0}) belongs to D_{ℂ−c}^{b}(k_{𝔛}) and its microsupport is contained in the normal cone C(Λ_{0}, Λ_{1}),

(ii) the natural morphism

RHom_{W_{𝔛}}(L_{1}, L_{0}) ⟶ D_{𝔛}(RHom_{W_{𝔛}}(L_{0}, L_{1} [dim_{ℂ} 𝔛]))

is an isomorphism.

The proof of (i) will be given in § 8 and (ii) is a particular case of Theorem 6.1.

Conjecture 7.5. Theorem 7.4 remains true without assuming that the Λ_{i}’s are smooth.

Remark that the analogous of Conjecture 7.5 for complex contact manifolds is true over the field ℂ, as we shall see in § 9.

Corollary 7.6. Let L_{0} and L_{1} be two regular holonomic W_{𝔛}-modules supported by smooth Lagrangian submanifolds. Then the object RHom_{W_{𝔛}}(L_{1}, L_{0}) of D_{ℂ−c}^{b}(k_{𝔛}) is perverse.

Proof. Since the problem is local, we may assume that 𝔛 = T^{∗}X, Λ_{0} = T_{X}^{∗}X and L_{0} = O_{X}^{τ}.

By Theorem 7.4 (ii), it is enough to check that if Y is a locally closed smooth submanifold of X of codimension l, then H^{j}(RΓ_{Y} (RHom_{W_{T^{∗}X}} (L_{1}, L_{0}))|_{Y} vanishes for j < l. This follows from Lemma 5.9. q.e.d.

Remark 7.7. It would be interesting to compare RHom_{W_{𝔛}}(L_{1}, L_{0}) with the complexes obtained in [1].

8 Proof of Theorem 7.4

In this section X denotes a complex manifold. As usual, O_{X} is the structure sheaf and D_{X} is the sheaf of rings of (finite–order) differential operators. For a coherent D_{X}-module M, we denote by char(M) its characteristic variety. This notion extends to the case where M is a countable union of coherent sub-D_{X}-modules. In this case, one sets

char(M) = \overline{⋃_{N ⊂M} char(N)}

where N ranges over the family of coherent D_{X}-submodules of M, and, for a subset S of T^{∗}X, \overline{S} means the closure of S.

Lemma 8.1. Let M_{0} be a coherent D_{X}-module. Then

SS(RHom_{D_{X}} (M_{0}, O_{X}^{τ}(0))) ⊂ char(M_{0}). (8.1)

Proof. Apply [19, Ch.3 Th. 3.2.1]. q.e.d.

Lemma 8.2. Let M be a coherent D_{X}[τ^{−1}]-module. Then

SS(RHom_{D_{X}[τ^{−1}]}(M, O_{X}^{τ}(0))) ⊂ char(M). (8.2)

Proof. Let M_{0} ⊂ M be a coherent D_{X}-submodule which generates M. Then

char(D_{X}[τ^{−1}] ⊗_{D_{X}} M_{0}) = char(M_{0}) ⊂ char(M).

Consider the exact sequence of coherent D_{X}[τ^{−1}]-modules

0 ⟶ N ⟶ D_{X}[τ^{−1}] ⊗_{D_{X}} M_{0} ⟶ M ⟶ 0. (8.3)

Applying the functor RHom_{D_{X}[τ^{−1}]}(^{•} , O_{X}^{τ}) to the exact sequence (8.3),

we get a distinguished triangle G′ ⟶ G ⟶ G′′ ⟶^{+1} . Note that G ≃ RHom_{D_{X}} (M_{0}, O_{X}^{τ}), since D_{X}[τ^{−1}] is flat over D_{X}.

Let θ = (x_{0}; p_{0}) ∈ T^{∗}X with θ ∉ char(M) and let ψ be a real function on X such that ψ(x_{0}) = 0 and dψ(x_{0}) = p_{0}. Consider the distinguished triangle

(RΓ_{ψ≥0}(G′))_{x_{0}} ⟶ (RΓ_{ψ≥0}(G))_{x_{0}} ⟶ (RΓ_{ψ≥0}(G′′))_{x_{0}} ⟶^{+1} . (8.4)

By Lemma 8.1, we have:

H^{j}((RΓ_{ψ≥0}(G))_{x_{0}}) ≃ 0 for all j ∈ ℤ.

The objects of the distinguished triangle (8.4) are concentrated in degree ≥ 0. Therefore, H^{j}((RΓ_{ψ≥0}(G′))_{x_{0}}) ≃ 0 for j ≤ 0.

Since char(N) ⊂ char(D_{X}[τ^{−1}]⊗_{D_{X}} M_{0}), we get H^{j}((RΓ_{ψ≥0}(G′′))_{x_{0}}) ≃ 0 for j ≤ 0. By repeating this argument, we deduce that H^{j}((RΓ_{ψ≥0}(G′))_{x_{0}}) ≃ H^{j−1}((RΓ_{ψ≥0}(G′′))_{x_{0}}) ≃ 0 for all j ∈ ℤ. q.e.d.

Note that the statement of Theorem 7.4 is local and invariant by quantized symplectic transformation. From now on, we denote by (x; u) a local symplectic coordinate system on 𝔛 such that

X := Λ_{0} = {(x; u) ∈ 𝔛; u = 0} .

We denote by (x; ξ) the associated homogeneous symplectic coordinates on T^{∗}X. The differential operator ∂_{x_{i}} on X has order 1 and principal symbol ξ_{i}. The monomorphism (1.7) extends as a monomorphism of rings

D_{X}[τ^{−1}, τ] ↪ W_{T^{∗}X}.

Note that the total symbol of the operator ∂_{x_{i}} of W_{T^{∗}X} is u_{i} · τ.

We may assume that there exists a holomorphic function φ: X ⟶ ℂ such that

Λ_{1} = {(x; u) ∈ 𝔛; u = grad φ(x)} . (8.5)

Here, grad φ = (φ′_{1}, …, φ′_{n}) and φ′_{i} = \frac{∂φ}{∂x_{i}}.

If Λ_{0} = Λ_{1}, Theorem 7.4 is immediate. We shall assume that Λ_{0} ≠ Λ_{1} and thus that φ is not a constant function. We may assume that

L_{0} = O_{X}^{τ},

L_{1} = W_{T^{∗}X}/I_{1}, I_{1} being the ideal generated by {∂_{x_{i}} −
φ′_{i}τ}_{i=1,...,n}.

(8.6)

To φ: X ⟶ ℂ are associated the maps

T^{∗}X ⟵^{φ_{d}} X ×_{ℂ} T^{∗}ℂ ⟶^{φ_{π}} T^{∗}ℂ,

and the (D_{X}, D_{ℂ})-bimodule D_{X⟶^{φ}ℂ}. Let t be the coordinate on ℂ. By identifying ∂_{t} and τ, we regard D_{X⟶^{φ}ℂ} as a D_{X}[τ]-module. We set

V = \overline{Im φ_{d}}, the closure of φ_{d}(T^{∗}ℂ ×_{ℂ} X).

Lemma 8.3. Regarding D_{X⟶^{φ}ℂ} as a D_{X}-module, one has char(D_{X⟶^{φ}ℂ}) ⊂ V.

Note that D_{X⟶^{φ}ℂ} is not a coherent D_{X}-module in general.

Proof. Let Γ_{φ} be the graph of φ in X × ℂ and let B_{φ} be the coherent left D_{X×ℂ}-module associated to this submanifold. (Note that char(B_{φ}) = Λ_{φ}, the conormal bundle to Γ_{φ} in T^{∗}(X × ℂ).) We shall identify Γ_{φ} with X by the first projection on X × ℂ and the left D_{X} modules B_{φ} with D_{X⟶^{φ}ℂ}.

Denote by δ(t − φ) the canonical generator of B_{φ}. Then

D_{X⟶^{φ}ℂ} = D_{X×ℂ} · δ(t − φ) = ∑_{n∈ℕ}D_{X}∂_{t}^{n} · δ(t − φ).

Define

N_{φ} := ∑_{n∈ℕ}D_{X}(t∂_{t})^{n} · δ(t − φ).

By [9, Th. 6.8], the D_{X}-module N_{φ} is coherent and its characteristic variety is contained in V. All D_{X}-submodules ∂_{t}^{n}N_{φ} of D_{X⟶^{φ}ℂ} are isomorphic since ∂_{t} is injective on B_{φ}. Then the result follows from

D_{X⟶^{φ}ℂ} = ∑_{n∈ℕ}∂_{t}^{n}N_{φ}.

q.e.d.

We consider the left ideal and the modules:

I:= the left ideal of D_{X}[τ^{−1}, τ] generated by {∂_{x_{i}} −φ′_{i}τ}_{i=1,...,n},
M = D_{X}[τ^{−1}]/(I ∩ D_{X}[τ^{−1}]),
M_{0} = D_{X}/(I ∩ D_{X}).

(8.7)

Lemma 8.4. (i) M is D_{X}[τ^{−1}]-coherent,

(ii) M_{0} is D_{X}-coherent,

(iii) we have an isomorphism of D_{X}-modules D_{X} · δ(t − φ) ≃ M_{0}, where D_{X} · δ(t − φ) is the D_{X}-submodule of D_{X⟶^{φ}ℂ} generated by δ(t − φ),

(iv) char(M) = char(M_{0}) ⊂ V.

Proof. (i) follows from Lemma 1.6 (ii). (ii) follows from Lemma 1.5 (iii). (iii) Clearly, D_{X} · δ(t − φ) ≃ D_{X}/D_{X} ∩ I. (iv) (a) By (iii) and Lemma 8.3 we get the inclusion char(M_{0}) ⊂ V. (iv) (b) Since M_{0} ⊂ M, the inclusion char(M_{0}) ⊂ char(M) is obvious. (iv) (c) Denote by u the canonical generator of M. Then M = ⋃_{n≤0} D_{X}τ^{n}u. Since there are epimorphisms M_{0}↠D_{X}τ^{n}u, the inclusion char(M) ⊂ char(M_{0}) follows. q.e.d.

Set

F_{0} = RHom_{D_{X}[τ^{−1}]}(M, O_{X}^{τ}(0)), (8.8)

where the module M is defined in (8.7).

Lemma 8.5. SS(F_{0}) is a closed ℂ^{×}-conic complex analytic Lagrangian subset of T^{∗}X contained in C(Λ_{0}, Λ_{1}).

Proof. By Lemmas 8.2 and 8.4, SS(F_{0}) ⊂ V ∩ (π^{−1}(Λ_{0} ∩ Λ_{1})), and one immediately checks that

V ∩ π^{−1}(Λ_{0} ∩ Λ_{1}) = C(Λ_{0}, Λ_{1}). (8.9)

Since SS(F_{0}) is involutive by [12] and is contained in a ℂ^{×}-conic analytic isotropic subset by Proposition 7.1, it is a closed ℂ^{×}-conic complex analytic Lagrangian subset of T^{∗}Λ_{0} by [12, Prop. 8.3.13]. q.e.d.

Lemma 8.6. Let F_{0} be as in (8.8). Then for each x ∈ X and each j ∈ ℤ, the k_{0}-module H^{j}(F_{0})_{x} is finitely generated.

Proof. Let x_{0} ∈ X and choose a local coordinate system around x_{0}. Denote by B(x_{0}; ε) the closed ball of center x_{0} and radius ε. By a result of [7] (see also [12, Prop. 8.3.12]), we deduce from Lemma 8.5 that the natural morphisms

RΓ(B(x_{0}; ε_{1}); F_{0}) ⟶ RΓ(B(x_{0}; ε_{0}); F_{0})

are isomorphisms for 0 < ε_{0} ≤ ε_{1} ≪ 1.

We represent F_{0} by a complex:

0 ⟶ (O^{τ} (0))^{N_{0}} ⟶^{d_{0}} ⋯ ⟶ (O^{τ} (0))^{N_{n}} ⟶ 0, (8.10)

where the differentials are k_{0}-linear. It follows from Lemma 5.5 and Theorem 4.3 that the cohomology objects H^{j}(F_{0})_{x_{0}} are finitely generated. q.e.d.

End of the proof of Theorem 7.4. As already mentioned, part (ii) is a particular case of Theorem 6.1.

Let us prove part (i). By “dévissage” we may assume that L_{0} and L_{1} are concentrated in degree 0. We may also assume that L_{0} and L_{1} are as in (8.6).

Let F_{0} be as in (8.8), and set F = RHom_{W_{T^{∗}X}} (L_{1}, O_{X}^{τ}). Since L_{1} ≃ W_{T^{∗}X} ⊗_{D_{X}[τ^{−1}]} M, we obtain

F ≃ F_{0} ⊗_{k_{0}} k

by Lemmas 1.6 and 1.7.

Hence we have SS(F) ⊂ SS(F_{0}), and the weak constructibility as well as the estimate of SS(F) follows by Lemma 8.2 and Lemma 8.5. The finiteness result follows from Lemma 8.6. q.e.d.

9 Serre functors on contact and symplectic manifolds

In the definition below, 𝕂 is a field and ^{⋆} denotes the duality functor for 𝕂-vector spaces.

Definition 9.1. Consider a 𝕂-triangulated category T .

(i) The category T is Ext -finite if for any L_{0}, L_{1} ∈ T , Ext_{T}^{j} (L_{1}, L_{0}) is finite-dimensional over 𝕂 for all j ∈ ℤ and is zero for |j| ≫ 0.

(ii) Assume that T is Ext -finite. A Serre functor (see [2]) S on T is an equivalence of 𝕂-triangulated categories S: T ⟶ T satisfying

(Hom_{T} (L_{1}, L_{0}))^{⋆} ≃ Hom_{T} (L_{0}, S(L_{1}))

functorially in L_{0}, L_{1} ∈ T .

(iii) If moreover there exists an integer d such that S is isomorphic to the shift by d, then one says that T is a 𝕂-triangulated Calabi-Yau category of dimension d.

Let 𝔜 be a complex contact manifold. The algebroid stack E_{𝔜} of microdifferential operators on 𝔜 has been constructed in [8] and the triangulated categories D_{coh}^{b}(E_{𝔜}), D_{hol}^{b}(E_{𝔜}) and D_{rh}^{b}(E_{𝔜}) are naturally defined.

Theorem 9.2. For a complex contact manifold 𝔜, we have

(i) for M and N in D_{rh}^{b}(E_{𝔜}), the object F = RHom_{E_{𝔜}}(M, N) belongs to D_{ℂ−c}^{b}(ℂ_{𝔜}),

(ii) if 𝔜 is compact, then D_{rh}^{b}(E_{𝔜}) is a Calabi-Yau ℂ-triangulated category of dimension dim_{ℂ} 𝔜 − 1.

Sketch of proof. (i) is well-known and follows from [10] (see [21] for further developments). The idea of the proof is as follows. The assertion being local and invariant by quantized contact transformations, we may assume that 𝔜 is an open subset of the projective cotangent bundle P^{∗}Y to a complex manifold. Then, using the diagonal procedure, we reduce to the case F = RHom_{D_{Y}} (M, C_{Z|Y}), where M is a regular holonomic D_{Y} -module and C_{Z|Y} is the E_{P^{∗}Y} -module associated to a complex hypersurface Z of Y. (ii) follows from (i) as in the proof of Theorem 6.1. q.e.d.

Remark 9.3. (i) If Conjecture 7.5 is true, that is, if Theorem 7.4 holds for any Lagrangian varieties, then, for any compact complex symplectic manifold 𝔛, the k-triangulated category D_{rh}^{b}(W_{𝔛}) is a Calabi-Yau k-triangulated category of dimension dim_{ℂ} 𝔛. Note that this result is true when replacing the notion of regular holonomic module by the notion of good modules, i.e., coherent modules admitting globally defined W_{𝔛}(0)-submodules which generate them. This follows from a theorem of Schapira-Schneiders to appear. (ii) Note that Proposition 1.4.8 of [10] is not true, but this proposition is not used in loc. cit. Indeed, for a compact complex manifold X, if T := D_{rh}^{b}(D_{X}) denotes the full triangulated subcategory of D^{b}(D_{X}) consisting of objects with regular holonomic cohomologies, it is well known that the duality functor is not a Serre functor on T . (iii) It may be interesting to notice that the duality functor is not a Serre functor in the setting of regular holonomic D-modules, but is a Serre functor in the microlocal setting, that is, for regular holonomic E-modules. This may be compared to [12, Prop. 8.4.14].

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Masaki Kashiwara Research Institute for Mathematical Sciences Kyoto University Kyoto, 606–8502, Japan e-mail: masaki@kurims.kyoto-u.ac.jp Pierre Schapira Institut de Mathématiques Université Pierre et Marie Curie 175, rue du Chevaleret, 75013 Paris, France e-mail: schapira@math.jussieu.fr http://www.math.jussieu.fr/∼schapira/