Quantization of complex Lagrangian submanifolds

Andrea D’Agnolo^{a,∗}

aUniversità di Padova - Dipartimento di Matematica via Trieste, 63 - 35121 Padova, Italy

Pierre Schapira^{b}

bUniversité Pierre et Marie Curie - Institut de Mathématiques 175, rue du Chevaleret - 75013 Paris, France

Abstract

Let Λ be a smooth Lagrangian submanifold of a complex symplectic manifold 𝔛. We construct twisted simple holonomic modules along Λ in the stack of deformationquantization modules on 𝔛.

Key words: quantization, microdifferential operators, complex Lagrangians, stacks 2000 MSC: 46L65, 14A20, 32C38

1 Introduction

Let 𝔜 be a complex contact manifold. A local model for 𝔜 is an open subset of the projective cotangent bundle P^{∗}Y to a complex manifold Y. The manifold P^{∗}Y is endowed with the sheaf E_{Y} of microdifferential operators of [20]. In [9], Kashiwara proves the existence of a canonical stack Mod(E_{𝔜}) on 𝔜, locally equivalent to the stack of E_{Y} -modules. Let Λ ⊂ 𝔜 be a smooth Lagrangian submanifold. In the same paper, Kashiwara states that there exists a globally

∗ Corresponding author.

Email addresses: dagnolo@math.unipd.it (Andrea D’Agnolo),

schapira@math.jussieu.fr (Pierre Schapira).
  URLs: www.math.unipd.it/∼dagnolo/ (Andrea D’Agnolo),
www.math.jussieu.fr/∼schapira/ (Pierre Schapira).

Preprint submitted to Elsevier 21 December 2006

defined holonomic system simple along Λ in the stack Mod(E_{𝔜}|_{Λ}) twisted by half-forms on Λ.

Now, let 𝔛 be a complex symplectic manifold. A local model for 𝔛 is an open subset of the cotangent bundle T^{∗}X to a complex manifold X. The manifold T^{∗}X is endowed with the sheaf W_{X} of WKB-differential operators, similar to E_{X}, but with an extra central parameter, a substitute to the lack of homogeneity. (Note that, in the literature, W_{X} is also called a deformationquantization ring, or a ring of semi-classical differential operators. See [19] for a precise description of the ring W_{X} and its links with E_{X}.) A stack Mod(W_{𝔛}) on 𝔛 locally equivalent to the stack of W_{X}-modules has been constructed in the formal case (in the general setting of Poisson manifolds) by [15] and in the analytic case (and by a different method, similar to [9]) by [19]. (See also [16,2,22] for papers closely related to this subject.)

In this paper, we prove that if Λ is a smooth Lagrangian submanifold of the complex symplectic manifold 𝔛, there exists a globally defined simple holonomic module along Λ in the stack Mod(W_{𝔛}|_{Λ}) twisted by half-forms on Λ. As a by-product, we prove that there is an equivalence of stacks between that of twisted regular holonomic modules along Λ and that of local systems on Λ. The local model for our theorem is given by 𝔛 = T^{∗}X and Λ = T_{X}^{∗}X, the zero-section of T^{∗}X. In this case, a simple module is 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 the family {f_{j}}_{j} satisfies certain growth conditions on compact subsets of X. The problem we solve here is how to patch together these local models.

Our proof consists in showing that if 𝔛 is a complex symplectic manifold and Λ a Lagrangian submanifold, then there exists a “contactification” 𝔜 of 𝔛 in a neighborhood of Λ. Local models for 𝔛 and 𝔜 are an open subset of the cotangent bundle T^{∗}X to a complex manifold X and an open subset of the projective cotangent bundle P^{∗}(X×ℂ), respectively. With the same techniques as in [9,19], we construct a stack Mod(E_{𝔜,t̂}) on 𝔜 locally equivalent to the stack of modules over the ring E_{X×ℂ,t̂} of microdifferential operators commuting with ∂/∂_{t}, where t is the coordinate on ℂ. We then apply Kashiwara’s existence theorem for simple modules along Lagrangian manifolds in the contact case to deduce the corresponding result in the symplectic case. In fact, the technical heart of this paper is devoted to giving a detailed proof, based on the theory of symbols of simple sections of holonomic modules, of Kashiwara’s result stated in [9].

Acknowledgements We would like to thank Louis Boutet de Monvel, Masaki Kashiwara, and Pietro Polesello for their useful comments and insights.

2 Stacks

Stacks were invented by Grothendieck and Giraud [6] and we refer to [14] for an exposition. Roughly speaking, a prestack (resp. a stack) is a presheaf (resp. a sheaf) of categories, as we shall see below.

In sections 2 and 3, we denote by X a topological space. However, all definitions and results easily extend when replacing X with a site, that is, a small category C_{X} endowed with a Grothendieck topology.

Definition 2.1. (a) A prestack S on X is the assignment of a category S(U) for every open subset U ⊂ X, a functor ρ_{V U}: S(U) ⟶ S(V) for every open inclusion V ⊂ U, and an isomorphism of functors λ_{W V U}: ρ_{W V} ρ_{V U} ⇒ ρ_{W U} for every open inclusion W ⊂ V ⊂ U, such that ρ_{UU} = id_{S(U)}, λ_{UUU} = id_{id_{S(U)}}, and the following diagram of isomorphisms of functors from S(U) to S(Y) commutes for every open inclusion Y ⊂ W ⊂ V ⊂ U

\xymatrix{ρ_{Y W} ρ_{W V} ρ_{V U} \ar@{=>}[r]^{λ_{Y W V} id_{ρ_{V U}}} \ar@{=>}[d]^{id_{ρ_{Y W}} λ_{W V U}} & ρ_{Y V} ρ_{V U} \ar@{=>}[d]^{λ_{Y V U}} \\ ρ_{Y W} ρ_{W U} \ar@{=>}[r]^{λ_{Y W U}} & ρ_{Y U}.}

For F ∈ S(U) and V ⊂ U, we will write F |_{V} for short instead of ρ_{V U}(F).

(b) A separated prestack is a prestack S such that for any F, G ∈ S(U), the presheaf Hom_{S|_{U}} (F, G), defined by V ↦ Hom_{S(V)}(F |_{V} , G|_{V}), is a sheaf.

A stack is a separated prestack satisfying suitable glueing conditions, which may be expressed in terms of descent data.

Definition 2.2. Let U be an open subset of X, U = {U_{i}}_{i∈I} an open covering of U and S a separated prestack on X.

(a) A descent datum on U for S is a pair

(\{F_{i}\}_{i∈I}, \{θ_{ij}\}_{i,j∈I}), with F_{i} ∈ S(U_{i}), θ_{ij}: F_{j}|_{U_{ij}} \underrightarrow{∼} F_{i}|_{U_{ij}} (2.1)

such that the following diagram of isomorphisms in S(U_{ijk}) commutes

\xymatrix{F_{j}|_{U_{ijk}} & F_{j}|_{U_{ij}}|_{U_{ijk}} \ar[l]_{λ} \ar[rr]^{θ_{ij}|_{U_{ijk}}} &  & F_{i}|_{U_{ij}}|_{U_{ijk}} \ar[r]^{λ} & F_{i}|_{U_{ijk}} \\ F_{j}|_{U_{jk}}|_{U_{ijk}} \ar[u]^{λ} &  &  &  & F_{i}|_{U_{ik}}|_{U_{ijk}} \ar[u]_{λ} \\  & F_{k}|_{U_{jk}}|_{U_{ijk}} \ar[r]|{λ} \ar[ul]^{θ_{jk}|_{U_{ijk}}} & F_{k}|_{U_{ijk}} & F_{k}|_{U_{ik}}|_{U_{ijk}}. \ar[ur]_{θ_{ik}|_{U_{ijk}}} \ar[l]|{λ}}

(b) The descent datum (2.1) is called effective if there exist F ∈ S(U) and isomorphisms θ_{i}: F|_{U_{i}} \underrightarrow{∼} F_{i} in S(U_{i}) satisfying the natural compatibility conditions with the θ_{ij}’s and λ’s.

Note that if the descent datum (2.1) is effective, then F is unique up to unique isomorphism.

Definition 2.3. A stack is a separated prestack such that for any open subset U of X and any open covering U = {U_{i}}_{i∈I} of U, the descent datum is effective.

To end this section, let us go up one level, and recall the glueing conditions for stacks.

Definition 2.4. Let U be an open subset of X, U = {U_{i}}_{i∈I} an open covering of U.

(a) A descent datum for stacks on U is a triplet

({S_{i}}_{i∈I}, {φ_{ij}}_{i,j∈I}, {α_{ijk}}_{i,j,k∈I}),  (2.2)

where the S_{i}’s are stacks on U_{i}, φ_{ij}: S_{j}|_{U_{ij}} ⟶ S_{i}|_{U_{ij}} are equivalences of stacks, and α_{ijk}: φ_{ij}φ_{jk} ⇒ φ_{ik}: S_{k}|_{U_{ijk}} ⟶ S_{i}|_{U_{ijk}} are isomorphisms of functors such that for any i, j, k, l ∈ I the following diagram of isomorphisms of functors from S_{l}|_{U_{ijkl}} to S_{i}|_{U_{ijkl}} commutes

\xymatrix{φ_{ij}φ_{jk}φ_{kl} \ar@{=>}[r]^{α_{ijk} id_{φ_{kl}}} \ar@{=>}[d]^{id_{φ_{ij}} α_{jkl}} & φ_{ik}φ_{kl} \ar@{=>}[d]^{α_{ikl}} \\ φ_{ij}φ_{jl} \ar@{=>}[r]^{α_{ijl}} & φ_{il}.}  (2.3)

(b) The descent datum (2.2) is called effective if there exist a stack S on U, equivalences of stacks φ_{i}: S|_{U_{i}} ⟶ S_{i} and isomorphisms of functors α_{ij}: φ_{ij}φ_{j} ⇒ φ_{i}: S|_{U_{ij}} ⟶ S_{i}|_{U_{ij}}, satisfying the natural compatibility conditions.

Note that if the descent datum (2.3) is effective, then S is unique up to equivalence and such an equivalence is unique up to unique isomorphism.

In the language of 2-categories, the following theorem asserts that the 2prestack of stacks is a 2-stack.

Theorem 2.5. (cf [6,3]) Descent data for stacks are effective.

Denote by S the stack associated with the descent datum (2.3). Objects of S(U) can be described by pairs

({F_{i}}_{i∈I}, {ξ_{ij}}_{i,j∈I}),  (2.4)

where F_{i} ∈ S_{i}(U_{i}) and ξ_{ij}: φ_{ij}(F_{j}|_{U_{ij}}) ⟶ F_{i}|_{U_{ij}} are isomorphisms in S_{i}(U_{ij}) such that for i, j, k ∈ I the following diagram in S_{i}(U_{ijk}) commutes

\xymatrix{φ_{ij}φ_{jk}(F_{k}|_{U_{ijk}}) \ar[r]^{φ_{ij}(ξ_{jk})} \ar[d]^{α_{ijk}(F_{k})} & φ_{ij}(F_{j}|_{U_{ijk}}) \ar[d]^{ξ_{ij}} \\ φ_{ik}(F_{k}|_{U_{ijk}}) \ar[r]^{ξ_{ik}} & F_{i}|_{U_{ijk}}.}  (2.5)

3 Twisted modules

Let us now recall how stacks of twisted modules are constructed in [8,9].

Let k be a commutative unital ring and A a sheaf of k-algebras on X. Denote by Mod(A) the category of left A-modules, and by Mod(A) the associated stack X ⊃ U ↦ Mod(A|_{U}).

Consider an open covering U = {U_{i}}_{i∈I} of X, a family of k-algebras A_{i} on U_{i} and k-algebra isomorphisms f_{ij}: A_{j}|_{U_{ij}} ⟶ A_{i}|_{U_{ij}}. The existence of a sheaf of k-algebras locally isomorphic to A_{i} requires the condition f_{ij}f_{jk} = f_{ik} on triple intersections. The weaker conditions (3.2) and (3.3) below are needed for the existence of a k-additive stack locally equivalent to Mod(A_{i}).

Definition 3.1. A k-algebroid descent datum A on U is a triplet

A = ({A_{i}}_{i∈I}, {f_{ij}}_{i,j∈I}, {a_{ijk}}_{i,j,k∈I}),  (3.1)

where A_{i} is a k-algebra on U_{i}, f_{ij}: A_{j}|_{U_{ij}} ⟶ A_{i}|_{U_{ij}} is a k-algebra isomorphism, a_{ijk} ∈ A_{i}^{×}(U_{ijk}) is an invertible section, and (3.2) and (3.3) below are satisfied:

f_{ij}f_{jk} = Ad(a_{ijk})f_{ik} as k-algebra isomorphisms A_{k}|_{U_{ijk}} \underrightarrow{∼} A_{i}|_{U_{ijk}}, (3.2)
a_{ijk}a_{ikl} = f_{ij}(a_{jkl})a_{ijl} in A_{i}^{×}(U_{ijkl}).  (3.3)

(Here Ad(a_{ijk}) denotes the automorphism of A_{i}|_{U_{ijk}} given by a ↦ a_{ijk} a a_{ijk}^{−1}.)

Remark 3.2. The notion of an algebroid stack exists intrinsically, without using coverings or descent data. It has been introduced by [15] and developed in [4]. In this paper, we shall restrict ourselves to algebroids presented by descent data.

Remark 3.3. Let B_{i} ⊂ A_{i}^{×} be multiplicative subgroups, invariant by f_{ij}, and such that for any b_{i}, b′_{i} ∈ B_{i}, the equality Ad(b_{i}) = Ad(b′_{i}) implies b_{i} = b′_{i}. Assume that a_{ijk} ∈ B_{i}. Then, as noticed e.g., in [9, pag. 2], condition (3.3) follows from (3.2).

Let us recall how to define the stack of “A-modules” in terms of local data.

A k-algebra morphism f: B ⟶ A induces a functor

f̃: Mod(A) ⟶ Mod(B)

defined by M ↦ _{f}M, where _{f}M denotes the sheaf of k-vector spaces M, endowed with the B-module structure given by bm := f(b)m for b ∈ B and m ∈ M.

For a ∈ A^{×} an invertible section, the automorphism Ad(a) induces the functor { \atop Ad(a)} between Mod(A) and itself, and we denote by

ã: { \atop Ad(a)} ⇒ id_{Mod(A)}

the isomorphism of functors given by ã(M): _{Ad(a)}M ⟶ M, u ↦ a^{−1}u, for u ∈ M ∈ Mod(A). (Note that ã(M)(a′u) = a^{−1}aa′a^{−1}u = a′ã(M)(u).)

Definition 3.4.(i) The stack of twisted modules associated to the k-algebroid descent datum A on U in (3.1) is the stack defined (using Theorem 2.5) by the descent datum

Mod(A) = ({Mod(A_{i})}_{i∈I}, {f̃_{ji}}_{i,j∈I}, {ã_{kji}}_{i,j,k∈I}).  (3.4)

(ii) One sets Mod(A) := Mod(A)(X). Objects of the category Mod(A) are called twisted modules.

According to (2.4), objects of Mod(A) are described by pairs

M = ({M_{i}}_{i∈I}, {ξ_{ij}}_{i,j∈I}),

where M_{i} are A_{i}-modules and ξ_{ij}: _{f_{ji}}M_{j}|_{U_{ij}} ⟶ M_{i}|_{U_{ij}} are isomorphisms of A_{i}-modules, such that for any u_{k} ∈ M_{k} one has

ξ_{ij}(_{f_{ji}}ξ_{jk}(u_{k})) = ξ_{ik}(a_{kji}^{−1}u_{k})  (3.5)

as morphisms _{f_{kj}f_{ji}}M_{k} ⟶ M_{i}. Indeed, (3.5) translates the commutativity of (2.5).

Example 3.5. Let X be a complex manifold, and denote by Ω_{X} the sheaf of holomorphic forms of maximal degree. Take an open covering U = {U_{i}}_{i∈I} of X such that there are nowhere vanishing sections ω_{i} ∈ Ω_{U_{i}}. Let t_{ij} ∈ O_{U_{ij}}^{×} be the transition functions given by ω_{j}|_{U_{ij}} = t_{ij}ω_{i}|_{U_{ij}}. Choose determinations s_{ij} ∈ O_{U_{ij}}^{×} for the multivalued functions t_{ij}^{1/2}. Since s_{ij}s_{jk} and s_{ik} are both determinations of t_{ik}^{1/2}, there exists c_{ijk} ∈ {−1, 1} such that

s_{ij}s_{jk} = c_{ijk}s_{ik}.  (3.6)

We thus get a ℂ-algebroid descent datum

ℂ\sqrt{Ω_{X}} := (\{ℂ_{U_{i}}\}_{i∈I}, \{id_{ℂ_{U_{ij}}}\}_{i,j∈I}, \{c_{ijk}\}_{i,j,k∈I}).

Note that, since c_{ijk}^{2} = 1, there is an equivalence Mod(ℂ\sqrt{Ω_{X}}) ≃ Mod(ℂ\sqrt{Ω_{X}^{−1}}).

Recall from [8] (see also [5, §1]), that there is an equivalence

Mod(ℂ\sqrt{Ω_{X}}) ≃ Mod(ℂ_{X})  (3.7)

if and only if the cohomology class [c_{ijk}] ∈ H^{2}(X; ℂ_{X}^{×}) is trivial. Consider the long exact cohomology sequence

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

associated with the short exact sequence

1 ⟶ ℂ_{X}^{×} ⟶ O_{X}^{×} ⟶^{d log} dO_{X} ⟶ 0.

One has [c_{ijk}] = γ(\frac{1}{2}β([Ω_{X}])), so that [c_{ijk}] = 1 if and only if there exists a line bundle L such that \frac{1}{2}β([Ω_{X}]) = β([L]), i.e. such that β([Ω_{X} ⊗_{O} L^{−2}]) = 0. This last condition holds if and only if there exists a local system of rank one L such that [Ω_{X} ⊗_{O} L^{−2}] = α([L]). Summarizing, (3.7) holds if and only if there exist L and L as above, such that

Ω_{X} ≃ L ⊗ L^{2}.

The twisted sheaf of half-forms in Mod(ℂ\sqrt{Ω_{X}}) is given by

\sqrt{Ω_{X}} = (\{O_{U_{i}}\}_{i∈I}, \{s_{ij}\}_{i,j∈I}).

We denote by \sqrt{ω_{i}} the section corresponding to 1 ∈ O_{U_{i}}. Hence, on U_{ij} we have

\sqrt{ω_{j}} = s_{ij}\sqrt{ω_{i}}.

Denote by [s_{ij}] the equivalence class of s_{ij} in O_{U_{ij}}^{×}/ℂ_{U_{ij}}^{×}. Since the s_{ij}’s satisfy (3.6), we notice that

\sqrt{Ω_{X}^{×}}/ℂ_{X}^{×} = (\{O_{U_{i}}^{×}/ℂ_{U_{i}}^{×}\}_{i∈I}, \{[s_{ij}]\}_{i,j∈I}) ∈ Mod(ℤ_{X})  (3.8)

is a (usual, i.e. not twisted) sheaf.

Let A = ({A_{i}}_{i∈I}, {f_{ij}}_{i,j∈I}, {a_{ijk}}_{i,j,k∈I}) be a k-algebroid descent datum on U as in (3.1), and consider a pair

M = ({M_{i}}_{i∈I}, {ξ_{ij}}_{i,j∈I}),

where M_{i} are A_{i}-modules and ξ_{ij}: _{f_{ji}}M_{j}|_{U_{ij}} ⟶ M_{i}|_{U_{ij}} are isomorphisms of A_{i}-modules which do not necessarily satisfy (3.5). Assume instead that there are isomorphisms

Hom_{A_{i}}(_{f_{ji}}M_{j}|_{U_{ij}}, M_{i}|_{U_{ij}}) ≃ k_{U_{ij}}.  (3.9)

Under this assumption, we will show in Proposition 3.8 below that M makes sense as a global object of Mod(A ⊗_{k} S), for a suitable twist S.

Consider the isomorphisms

ϕ_{ij}: Hom_{A_{i}}(_{f_{ji}}M_{j}|_{U_{ij}}, M_{i}|_{U_{ij}}) \underrightarrow{∼} k_{U_{ij}},  (3.10)

defined by ξ_{ij} ↦ 1. The dotted arrow defined by the following commutative diagram of isomorphisms is the multiplication by a section c_{ijk} of k_{U_{ijk}}^{×}.

\xymatrix{Hom_{A_{j}}(_{f_{ij}}M_{i}, M_{j}) ⊗_{k} Hom_{A_{k}}(_{f_{jk}}M_{j}, M_{k}) \ar[r]_{∼}^{φ_{ji}⊗φ_{kj}} \ar[d]_{≀}^{_{_{f_{jk}}⊗id}} & k_{U_{ijk}} ⊗_{k} k_{U_{ijk}} \ar[d]|{≀} \\ Hom_{A_{k}}(_{f_{ij}f_{jk}}M_{i},_{f_{jk}}M_{j}) ⊗_{k} Hom_{A_{k}}(_{f_{jk}}M_{j}, M_{k}) & k_{U_{ijk}} \\ Hom_{A_{k}}(_{f_{ik}}M_{i},_{f_{jk}}M_{j}) ⊗_{k} Hom_{A_{k}}(_{f_{jk}}M_{j}, M_{k}) \ar[d]_{≀}^{∘} \ar[u]^{≀}_{_{_{a_{ijk}}⊗id}} \\ Hom_{A_{k}}(_{f_{ik}}M_{i}, M_{k}) \ar[r]_{∼}^{φ_{ki}} & k_{U_{ijk}}. \ar@{.>}[uu]_{≀}^{c_{ijk}}} (3.11)

Here, the first vertical arrow on the left follows from the equality _{f_{ij}f_{jk}}M_{i} = f_{jk}(_{f_{ij}}M_{i}), and the second one from _{f_{ij}f_{jk}}M_{i} = _{Ad(a_{ijk})f_{ik}}M_{i}.

Lemma 3.6. The constants c_{ijk} defined above satisfy the cocycle condition

c_{ijk}c_{ikl} = c_{jkl}c_{ijl}.

Proof. Consider morphisms

η_{ji}: _{f_{ij}}M_{i} ⟶ M_{j}.

These induce morphisms

f_{jk}η_{ji}: _{f_{jk}}(_{f_{ij}}M_{i}) ⟶ _{f_{jk}}M_{j}.

The composition

f_{ik}M_{i} ⟶^{_{a_{ijk}}^{−1}} _{Ad(a_{ijk})f_{ik}}M_{i} = _{f_{jk}}(_{f_{ij}}M_{i}) ⟶^{_{f_{jk}} η_{ji}} _{f_{jk}}M_{j}

is given by

u_{i} ↦ a_{ijk}u_{i} ↦ _{f_{jk}}η_{ji}(a_{ijk}u_{i}).

Hence, the composition of the vertical isomorphisms in the left column of (3.11) is given by

η_{ji} ⊗ η_{kj} ↦ η_{kj}(_{f_{jk}}η_{ji}(a_{ijk} · ^{•})).

One then has

φ_{ji}(η_{ji})φ_{kj}(η_{kj}) = φ_{ki}(η_{kj}(_{f_{jk}}η_{ji}(a_{ijk} · ^{•})))c_{ijk}.  (3.12)

Using (3.12), we have, on one hand

φ_{ji}(η_{ji})φ_{kj}(η_{kj})φ_{lk}(η_{lk}) = φ_{ji}(η_{ji})φ_{lj}(η_{lk}(_{f_{kl}}η_{kj}(a_{jkl} · ^{•})))c_{jkl}
= φ_{li}(η_{lk}(_{f_{kl}}η_{kj}(a_{jkl}_{f_{jl}}η_{ji}(a_{ijl} · ^{•}))))c_{ijl}c_{jkl}

= φ_{li}(η_{lk}(_{f_{kl}}η_{kj}(_{f_{jk}f_{kl}}η_{ji}(f_{ij}(a_{jkl})a_{ijl} · ^{•}))))c_{ijl}c_{jkl},

and on the other hand

φ_{ji}(η_{ji})φ_{kj}(η_{kj})φ_{lk}(η_{lk}) = φ_{ki}(η_{kj}(_{f_{jk}}η_{ji}(a_{ijk} · ^{•})))φ_{lk}(η_{lk})c_{ijk}
= φ_{li}(η_{lk}(_{f_{kl}}η_{kj}(_{f_{jk}f_{kl}}η_{ji}(a_{ikl}a_{ijk} · ^{•}))))c_{ikl}c_{ijk}.

The conclusion follows using (3.3). q.e.d.

Remark 3.7. Lemma 3.6 is a particular case of a general result which asserts that equivalence classes of locally trivial k-algebroids on X are in one-to-one correspondence with H^{2}(X; k^{×}). The analogue result for gerbes is discussed in [6], and we refer to [4] for the formulation in terms of algebroids.

Let us recall our setting. Given a k-algebroid descent datum

A = ({A_{i}}_{i∈I}, {f_{ij}}_{i,j∈I}, {a_{ijk}}_{i,j,k∈I}),

consider a pair

M = ({M_{i}}_{i∈I}, {ξ_{ij}}_{i,j∈I}),

where M_{i} are A_{i}-modules and ξ_{ij}: _{f_{ji}}M_{j}|_{U_{ij}} ⟶ M_{i}|_{U_{ij}} are isomorphisms of A_{i}-modules. Assuming (3.9), Lemma 3.6 guarantees that the constants c_{ijk} defined by (3.11) satisfy the cocycle condition. We can thus consider the kalgebroid descent datum

S = ({k_{U_{i}}}_{i∈I}, {id_{k_{U_{ij}}} }_{i,j∈I}, {c_{ijk}}_{i,j,k∈I}).

Set

A ⊗_{k} S = ({A_{i}}_{i∈I}, {f_{ij}}_{i,j∈I}, {a_{ijk}c_{ijk}}_{i,j,k∈I}).

Proposition 3.8. Let M = ({M_{i}}_{i∈I}, {ξ_{ij}}_{i,j∈I}) be as above, and assume (3.9), then

M = (M_{i}, ξ_{ij}) ∈ Mod(A ⊗_{k} S).

Proof. By (3.5), it is enough to show that

ξ_{ik} = ξ_{ij}(_{f_{ji}}ξ_{jk}(a_{kji}c_{kji} · ^{•})).

We have φ_{ik}(ξ_{ik}) = 1 = φ_{jk}(ξ_{jk})φ_{ij}(ξ_{ij}) = φ_{ik}(ξ_{ij}(_{f_{ji}}ξ_{jk}(a_{kji}c_{kji} · ^{•}))) by (3.12). The conclusion follows since φ_{ik} is an isomorphism. q.e.d.

4 Microdifferential modules

Here we review a few notions from the theory of microdifferential modules. References are made to [20] and also to [12] for complementary results. See [10,21] for an exposition.

Let Y be a complex analytic manifold, and π: T^{∗}Y ⟶ Y its cotangent bundle. The sheaf E_{Y} of microdifferential operators on T^{∗}Y is a ℂ-central algebra endowed with a ℤ-filtration by the order. Denote by E_{Y} (m) its subsheaf of operators of order at most m, and by O_{T^{∗}Y} (m) the sheaf of functions homogeneous of degree m in the fiber of π. Denote by eu the Euler vector field, i.e. the infinitesimal generator of the action of ℂ^{×} on T^{∗}Y. Then f ∈ O_{T^{∗}Y} (m) if and only if eu f = mf. In a homogeneous symplectic local coordinate system (x; ξ) on U ⊂ T^{∗}Y, a section P ∈ Γ(U; E_{Y} (m)) is written as a formal series

P = ∑_{j≤m} p_{j}(x; ξ), p_{j} ∈ Γ(U; O_{T^{∗}Y} (j)),  (4.1)

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

If Q = ∑_{j≤n} q_{j} is another section, the product P Q = R = ∑_{j≤m+n} r_{j} is given by the Leibniz rule

r_{k} = ∑_{k=i+j−|α|} \frac{1}{α!}(∂_{ξ}^{α}p_{i})(∂_{x}^{α}q_{j}).

The symbol map

σ_{m}: E_{Y} (m) ⟶ O_{T^{∗}Y} (m), P ↦ p_{m}

does not depend on the choice of coordinates and induces the symbol map

σ: E_{Y} ⟶ gr E_{Y} \underrightarrow{∼} ⨁_{m∈ℤ}O_{T^{∗}Y} (m).

The formal adjoint of P = ∑_{j≤m} p_{j} is defined by

P^{∗} = ∑_{j≤m} p_{j}^{∗}, p_{j}^{∗}(x; ξ) = ∑_{j=k−|α|} \frac{(−1)^{|α|}}{α!} ∂_{ξ}^{α}∂_{x}^{α}p_{k}(x; −ξ).

It depends on the choice of coordinates, and more precisely on the choice of the top degree form dx_{i} ∧ ⋯ ∧ dx_{n} ∈ Ω_{Y}. One thus considers the twist of E_{Y} by half-forms

E_{Y}^{√\overline{v}} := π^{−1}\sqrt{Ω_{Y}} ⊗_{π^{−1}O_{Y}} E_{Y} ⊗_{π^{−1}O_{Y}} π^{−1}\sqrt{Ω_{Y}^{−1}}.

This is a sheaf of filtered ℂ-algebras endowed with a canonical anti-isomorphism

∗: E_{Y}^{√\overline{v}} \underrightarrow{∼} a_{∗}E_{Y}^{√\overline{v}},

where a denotes the antipodal map on T^{∗}Y. There is a subprincipal symbol

σ′_{m−1} : E_{Y}^{√\overline{v}}(m) ⟶ O_{T^{∗}Y} (m − 1), P ↦ \frac{1}{2}σ_{m−1}(P − (−1)^{m}P^{∗}).

In local coordinates, σ′_{m−1}(P) = p_{m−1} − \frac{1}{2} ∑_{i} ∂_{x_{i}}∂_{ξ_{i}}p_{m} for P as in (4.1).

The following definition is adapted from [12].

Definition 4.1. Let Λ be a smooth, locally closed, ℂ^{×}-conic submanifold of T^{∗}Y. Let M be a coherent E_{Y} -module supported by Λ.

(a) One says that M is regular (resp. simple) along Λ if there locally exists a coherent sub-E_{Y} (0)-module M_{0} of M which generates it over E_{Y}, and such that M_{0}/E_{Y} (−1)M_{0} is an O_{Λ}(0)-module (resp. a locally free O_{Λ}(0)- module of rank one).

(b) Let M be simple along Λ. A section u ∈ M is called a simple generator if M_{0} = E_{Y} (0)u satisfies the conditions in (a), and the image of u in M_{0}/E_{Y} (−1)M_{0} generates this module over O_{Λ}(0).

Set

I_{Λ} = {P ∈ E_{Y} (1)|_{Λ}; σ_{1}(P)|_{Λ} = 0},  (4.2)

and denote by E_{Λ} the sub-algebra of E_{Y} |_{Λ} generated by I_{Λ}.

Remark 4.2. Let u be a generator of a coherent E_{Y} -module M. Then M ≃ E_{Y} /I, where I = {P ∈ E_{Y} : P u = 0}. Set

\overline{I} = \{σ_{m}(P); m ∈ ℤ, P ∈ I ∩ E_{Y} (m)\},

and note that supp M = supp(O_{T^{∗}Y} /\overline{I}). Then M is simple if and only if there locally exists a generator u such that the ideal \overline{I} is reduced. Moreover, such a section u is a simple generator and the sub-E_{Y} (0)-module M_{0} generated by u satisfies

E_{Λ}M_{0} ⊂ M_{0}.

Indeed, in a homogeneous symplectic local coordinate system we may write P ∈ I_{Λ} as P = P′ + Q with Q of order ≤ 0 and P′u = 0.

Symbol of sections of simple systems

Let us recall the notion of symbol for simple generators. References are made to [20,7].

For a vector field v ∈ Θ_{Λ} on Λ, denote by L_{v}^{1/2} its Lie derivative action on the twisted sheaf \sqrt{Ω_{Λ}}. Then L_{v}^{1/2} is an operator of order one in the ring D_{Λ}^{√\overline{v}} = \sqrt{Ω_{Λ}} ⊗_{O_{Λ}} D_{Λ} ⊗_{O_{Λ}} \sqrt{Ω_{Λ}^{−1}}

of differential operators acting on \sqrt{Ω_{Λ}}.

Define I_{Λ}^{√\overline{v}} and E_{Λ}^{√\overline{v}} as in (4.2), replacing E_{Y} with E_{Y}^{√\overline{v}}, and denote by H_{f} the Hamiltonian vector field of f ∈ O_{T^{∗}Y}. Note that H_{f} ∈ T Λ if f|_{Λ} = 0. For P ∈ I_{Λ}^{√\overline{v}}, consider the transport operator

L(P) = L_{H_{σ_{1}(P)}|_{Λ}}^{1/2} +σ′_{0}(P)|_{Λ}.

One checks that L satisfies the relations L(AP) = σ_{0}(A)L(P), L(P A) = L(P)σ_{0}(A), and L([P, Q]) = [L(P), L(Q)], for P, Q ∈ I_{Λ}^{√\overline{v}} and A ∈ E_{Λ}^{√\overline{v}}(0) (see e.g. [10, §8.3]). It follows that L extends as a ℂ-algebra morphism

L: E_{Λ}^{√\overline{v}} ⟶ D_{Λ}^{√\overline{v}}  (4.3)

by setting L(P_{1} ⋯ P_{r}) = L(P_{1}) ⋯ L(P_{r}), for P_{i} ∈ I_{Λ}^{√\overline{v}}.

Let M be a simple E_{Y}^{√\overline{v}}-module along Λ, and u ∈ M a simple generator. The twisted subsheaf of \sqrt{Ω_{Λ}} defined by

\{σ ∈ \sqrt{Ω_{Λ}}; L(P)σ = 0 ∀P ∈ E_{Λ}^{√\overline{v}}, P u = 0\}  (4.4)

is locally a free sheaf of rank one over ℂ.

Definition 4.3. Let M be a simple E_{Y}^{√\overline{v}}-module along Λ and let u ∈ M be a simple generator. The symbol of u is defined by

σ_{Λ}(u) = [σ] ∈ \sqrt{Ω_{Λ}^{×}}/ℂ_{Λ}^{×}.

for σ as in (4.4).

The Euler vector field eu acts on \sqrt{Ω_{Λ}} by L_{eu}^{1/2}, and one says that σ ∈ \sqrt{Ω_{Λ}} is homogeneous of degree λ if eu σ = λσ. Hence, the notion of homogeneous section of \sqrt{Ω_{Λ}^{×}}/ℂ_{Λ}^{×} makes sense. One calls order of u the homogeneous degree of σ_{Λ}(u). Then the equivalence class of λ in ℂ/ℤ does not depend on u, and is called the order of M.

If P ∈ E_{Y}^{√\overline{v}}(m) is such that σ_{m}(P)|_{Λ} never vanishes, then

σ_{Λ}(P u) = σ_{m}(P)σ_{Λ}(u).  (4.5)

Also recall from loc.cit. that simple modules of the same order are locally isomorphic.

5 Quantization of contact manifolds

Here we review Kashiwara’s construction [9] of the stack of microdifferential modules on a contact manifold.

Let Y be a complex analytic manifold, π: T^{∗}Y ⟶ Y its cotangent bundle, Ṫ^{∗}Y = T^{∗}Y \ Y the complementary of the zero-section, ϖ: P^{∗}Y ⟶ Y the projective cotangent bundle, and γ: Ṫ^{∗}Y ⟶ P^{∗}Y the projection. The sheaf of microdifferential operators on P^{∗}Y is given by γ_{∗}(E_{Y} |_{Ṫ^{∗}X}), and we still denote it by E_{Y} for short. Since the antipodal map induces the identity on P^{∗}Y, the anti-involution ∗ is well defined on the sheaf E_{Y}^{√\overline{v}}.

Let χ_{ij}: P^{∗}Y_{i} ⊃ V_{i} ⟶ V_{j} ⊂ P^{∗}Y_{j} be a contact transformation. Recall that there locally exists a quantized contact transformation (QCT for short) above χ_{ij}. This is an isomorphism of filtered ℂ-algebras Φ_{ij}: χ_{ij}^{−1}E_{Y_{j}}^{√\overline{v}}|_{V_{j}} \underrightarrow{∼} E_{Y_{i}}^{√\overline{v}}|_{V_{i}}. Moreover, one can ask that Φ_{ij} is ∗-preserving. Such a quantization is not unique, but a key remark by Kashiwara is that ∗-preserving filtered automorphisms of E_{Y}^{√\overline{v}} are of the form Ad(Q) for a unique operator Q ∈ E_{Y}^{√\overline{v}}(0) satisfying

QQ^{∗} = 1, σ_{0}(Q) = 1.  (5.1)

Definition 5.1. A complex contact manifold 𝔜 = (𝔜, O_{𝔜}(1), α) is a complex manifold 𝔜 of dimension 2n + 1 endowed with a line bundle O_{𝔜}(1) and a 1form α ∈ Γ(𝔜, Ω_{𝔜}^{1}⊗_{O} O_{𝔜}(1)), such that α∧(dα)^{∧n} is a non-degenerate section of Ω_{𝔜}^{2n+1} ⊗_{O} O_{𝔜}(n + 1).

(Here we set O_{𝔜}(k) = O_{𝔜}(1)^{⊗k}, and we use the fact that α ∧ (dα)^{∧r} is a well-defined section of Ω_{𝔜}^{2r+1} ⊗_{O} O_{𝔜}(r + 1) for 0 ≤ r ≤ n.)

There is an open covering V = {V_{i}}_{i∈I} of 𝔜 and contact embeddings χ_{i}: V_{i} ↪ P^{∗}Y_{i}. Up to refining the covering (we still denote it by V), the induced contact transformations χ_{ij}: χ_{i}(V_{ij}) ⟶ χ_{j}(V_{ij}) can be quantized to a ∗-preserving filtered ℂ-algebra isomorphism

Φ_{ij}: χ_{ij}^{−1}(E_{Y_{j}}^{√\overline{v}}|_{χ_{j}(V_{ij})}) ⟶ E_{Y_{i}}^{√\overline{v}}|_{χ_{i}(V_{ij})}.

The composition Φ_{ij}Φ_{jk}Φ_{ik}^{−1} is a ∗-preserving automorphism of E_{Y_{i}}^{√\overline{v}}|_{χ_{i}(V_{ijk})}, and hence is equal to Ad(Q_{ijk}) for a unique Q_{ijk} ∈ E_{Y_{i}}^{√\overline{v}}(V_{ijk}) satisfying (5.1). This proves the theorem below, thanks to Remark 3.3.

Theorem 5.2. (cf [9, Theorem 2]) The triplet

E_{𝔜} = (\{χ_{i}^{−1}E_{Y_{i}}^{√\overline{v}}|_{V_{i}}\}_{i∈I}, \{χ_{i}^{−1}(Φ_{ij})\}_{i,j∈I}, \{χ_{i}^{−1}(Q_{ijk})\}_{i,j,k∈I})  (5.2)

is a ℂ-algebroid descent datum over 𝔜. In particular, there is an associated stack Mod(E_{𝔜}) on 𝔜 locally equivalent to the stack Mod(E_{Y_{i}}^{√\overline{v}}) of microdifferential modules.

Good modules

Any local notion, such as that of being coherent, simple, or regular, immediately extends to the category Mod(E_{𝔜}). Here, we will discuss the non local notion of being good (cf [7]).

Remark first that the construction in Theorem 5.2 also applies when replacing E_{Y_{i}} with E_{Y_{i}}(0). One thus gets a ℂ-algebroid descent datum E_{𝔜}(0) as well as a ℂ-linear functor E_{𝔜}(0) ⟶ E_{𝔜}. This induces a forgetful functor

Mod(E_{𝔜}) ⟶^{for} Mod(E_{𝔜}(0)).

This functor locally admits a left adjoint, hence it has a left adjoint

Mod(E_{𝔜}(0)) ⟶^{ext} Mod(E_{𝔜}).

Definition 5.3. A coherent module M ∈ Mod(E_{𝔜}) is good if for any relatively compact open subset U ⊂ 𝔜 there exists M_{0} ∈ Mod(E_{𝔜}(0)|_{U}) such that M|_{U} ≃ ext(M_{0}).

Quantization with parameters

Assume now that the bundle O_{𝔜}(1) is trivial, i.e. that there exists a nowhere vanishing section τ ∈ Γ(𝔜; O_{𝔜}(1)). Consider an open covering {V_{i}}_{i∈I} of 𝔜 and contact embeddings χ_{i}: V_{i} ↪ P^{∗}Y_{i} to which is attached the ℂ-algebroid descent datum (5.2).

Lemma 5.4. Up to refining the covering, there exist ∗-preserving QCT’s Φ_{ij} above χ_{ij}, and sections T_{i} ∈ Γ(χ_{i}(V_{i}); E_{Y_{i}}^{√\overline{v}}(1)), such that

σ(T_{i}) ∘ χ_{i} = τ, T_{i}^{∗} = −T_{i}, Φ_{ij}(T_{j}) = T_{i}.

Proof. Let (t, x_{1}, …, x_{n}, u_{1}, …, u_{n}) be a local coordinate system on V_{i} such that the contact form is given by dt + ∑_{i} u_{i}dx_{i} and τ = σ(∂_{t}) ∘ χ_{i}. Then set T_{i} = ∂_{t}.

Let Φ_{ij}^{†} be a QCT above χ_{ij} such that Φ_{ij}^{†}(T_{j}) = T_{i}. Then the proof goes as that of [19, Lemma 5.3 (iii)]. Consider the QCT above the identity given by

Φ_{ij}^{‡} := Φ_{ij}^{†−1} ∘ ∗ ∘ Φ_{ij}^{†} ∘ ∗.

There exists Q ∈ E_{Y_{j}}^{√\overline{v}} such that Φ_{ij}^{‡} = Ad(QQ^{∗}). Hence Φ_{ij} := Φ_{ij}^{†} ∘ Ad(Q) is ∗-preserving QCT above χ_{ij}, and Φ_{ij}(T_{j}) = T_{i}. q.e.d.

Denote by E_{Y_{i},T_{i}}^{√\overline{v}} the subalgebra of E_{Y_{i}}^{√\overline{v}} of operators commuting with T_{i}. As above, denote by Q_{ijk} ∈ E_{Y_{i}}^{√\overline{v}}(χ_{i}(V_{ijk})) the unique operator satisfying (5.1) such that Φ_{ij}Φ_{jk}Φ_{ik}^{−1} = Ad(Q_{ijk}). Since Ad(Q_{ijk})(T_{i}) = T_{i}, it follows that Q_{ijk} is a section of E_{Y_{i},T_{i}}^{√\overline{v}}.

Proposition 5.5. Let 𝔜 be a complex contact manifold. Assume that there exists a nowhere vanishing section τ ∈ Γ(𝔜; O_{𝔜}(1)). Then the triplet

E_{𝔜,τ} = (\{χ_{i}^{−1}E_{Y_{i},T_{i}}^{√\overline{v}}|_{χ_{i}(V_{i})}\}_{i∈I}, \{χ_{i}^{−1}(Φ_{ij})\}_{i,j∈I}, \{χ_{i}^{−1}(Q_{ijk})\}_{i,j,k∈I})  (5.3)

is a ℂ-algebroid descent datum on V. In particular, there is an associated stack Mod(E_{𝔜,τ}) on 𝔜 locally equivalent to the stack Mod(E_{Y_{i},T_{i}}^{√\overline{v}}).

6 Simple holonomic modules on contact manifolds

Here, we give a proof of a result of Kashiwara [9] on the existence of twisted simple holonomic modules along smooth Lagrangian submanifolds (also called Legendrian in the literature) of complex contact manifolds.

Let 𝔜 be a complex contact manifold. Recall from Theorem 5.2 that there is an algebroid descent datum E_{𝔜} of microdifferential operators on 𝔜. Let Λ ⊂ 𝔜 be a Lagrangian submanifold. With notations as in Example 3.5, consider the stack Mod(E_{𝔜}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}) of twisted microdifferential modules on Λ.

Let us recall Kashiwara’s theorem announced in [9].

Theorem 6.1. Let 𝔜 be a complex contact manifold and let Λ ⊂ 𝔜 be a Lagrangian submanifold. There exists L ∈ Mod(E_{𝔜}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}) which is simple along Λ.

Remark 6.2. It will follow from the proof that moreover L is good.

Here, we give a proof of this result using the notion of symbol for simple sections of holonomic modules recalled in Section 4.

Proof. Let us denote for short by

E_{𝔜} = ({A_{i}}_{i∈I}, {f_{ij}}_{i,j∈I}, {a_{ijk}}_{i,j,k∈I})

the ℂ-algebroid descent datum (5.2) attached to an open covering 𝔜 = ⋃_{i∈I} V_{i}. Up to a refinement, we may assume that the ℂ-algebroid descent datum ℂ\sqrt{Ω_{Λ}} is attached to the same covering. More precisely, using notations as in Example 3.5 for X = Λ and U_{i} = Λ_{i} = Λ ∩ V_{i}, we assume that there are nowhere vanishing sections ω_{i} ∈ Ω_{Λ_{i}}, and s_{ij} ∈ O_{Λ_{ij}}^{×} with \sqrt{ω_{j}} = s_{ij}\sqrt{ω_{i}}, such that

ℂ\sqrt{Ω_{X}} = (\{ℂ_{Λ_{i}}\}_{i∈I}, \{id_{ℂ_{Λ_{ij}}}\}_{i,j∈I}, \{c_{ijk}\}_{i,j,k∈I}),

where the c_{ijk}’s are defined by

s_{ij}s_{jk} = c_{ijk}s_{ik}.

Up to a further refinement of the covering, we may assume that there exist simple A_{i}-modules L_{i} of order 0 along Λ_{i} and simple generators u_{i} of L_{i} such that

σ_{Λ_{i}}(u_{i}) = [\sqrt{ω_{i}}] ∈ (\sqrt{Ω_{Λ}^{×}}/ℂ_{Λ}^{×})|_{Λ_{i}}.

Since L_{i} and _{f_{ji}}L_{j} are simple A_{i}-modules along Λ_{ij} of the same order, they are isomorphic, and one has

Hom_{A_{i}}(_{f_{ji}}L_{j}, L_{i}) ≃ ℂ_{Λ_{ij}}.

Let us describe explicitly such an isomorphism. For ξ̃_{ij} ∈ Hom_{A_{i}}(_{f_{ji}}L_{j}, L_{i}), let b̃_{ij} is a section of A_{i} satisfying ξ̃_{ij}(u_{j}) = b̃_{ij}u_{i}. One has

σ_{Λ}(ξ̃_{ij}(u_{j})) = σ_{Λ}(u_{j}) = [\sqrt{ω_{j}}] = [s_{ij}\sqrt{ω_{i}}],
σ_{Λ}(b̃_{ij}u_{i}) = σ(b̃_{ij})σ_{Λ}(u_{i}) = [σ(b̃_{ij})\sqrt{ω_{i}}],

where the fourth equality follows from (4.5). In particular, s_{ij}^{−1}σ(b̃_{ij}) ∈ ℂ_{Λ_{ij}}^{×}. We can thus consider the isomorphism

φ_{ij}: Hom_{A_{i}}(_{f_{ji}}L_{j}, L_{i}) \underrightarrow{∼} ℂ_{Λ_{ij}}
ξ̃_{ij} ↦ s_{ij}^{−1}σ(b̃_{ij}).

Set

ξ_{ij} = φ_{ij}^{−1}(1): _{f_{ji}}L_{j} \underrightarrow{∼} L_{i},

and let b_{ij} ∈ A_{i} satisfy ξ_{ij}(u_{j}) = b_{ij}u_{i}. As φ_{ij}(ξ_{ij}) = 1, one has

σ(b_{ij}) = s_{ij}.  (6.1)

By Proposition 3.8, we get a twisted module

L = (L_{i}, ξ_{ij}) ∈ Mod(E_{𝔜}|_{Λ} ⊗_{ℂ} S),

where

S = ({ℂ_{Λ_{i}}}_{i∈I}, {id_{ℂ_{Λ_{ij}}}}_{i,j∈I}, {c̃_{ijk}}_{i,j,k∈I}),

the c̃_{ijk}’s being given by (3.11). Since φ_{ij}(ξ_{ij}) = 1 = φ_{jk}(ξ_{jk}), by (3.12) we have

c̃_{ijk} = φ_{ik}(ξ_{ij}(_{f_{ji}}ξ_{jk}(a_{kji} · ^{•}))).

To show that S = ℂ\sqrt{Ω_{Λ}}, it thus remains to prove that c̃_{ijk} = c_{ijk}, i.e. that

s_{ij}s_{jk} = c̃_{ijk}s_{ik}.

One has

ξ_{ij}(_{f_{ji}}ξ_{jk}(a_{kji}u_{k})) = f_{ij}f_{jk}(a_{kji}) ξ_{ij}(_{f_{ji}}ξ_{jk}(u_{k}))
= f_{ij}f_{jk}(a_{kji}) ξ_{ij}(b_{jk}u_{j})
= f_{ij}f_{jk}(a_{kji})f_{ij}(b_{jk}) ξ_{ij}(u_{j})
= f_{ij}f_{jk}(a_{kji})f_{ij}(b_{jk})b_{ij} u_{i}.

Then

c̃_{ijk} = φ_{ik}(ξ_{ij}(_{f_{ji}}ξ_{jk}(a_{kji} · ^{•})))
= s_{ik}^{−1}σ(f_{ij}f_{jk}(a_{kji})f_{ij}(b_{jk})b_{ij})
= s_{ik}^{−1}σ(b_{jk})σ(b_{ij})
= s_{ik}^{−1}s_{jk}s_{ij},

where the third equality follows from the fact that σ(a_{kji}) = 1, and the fourth one from (6.1). q.e.d.

Definition 6.3. A complex homogeneous symplectic manifold ℨ = (ℨ, ω, v) is a complex symplectic manifold (ℨ, ω) endowed with a vector field v satisfying L_{v} ω = ω.

Corollary 6.4 below was announced in [9]. Although we shall not use it, we give a proof for the reader’s convenience. Also note that our statement corrects that in loc. cit., following a private communication with M. Kashiwara.

Let 𝔜 = (𝔜, O_{𝔜}(1), α) be a contact manifold and denote by γ: 𝔜̃ ⟶ 𝔜 the total space of the ℂ^{×}-principal bundle associated with the dual of the line bundle O_{𝔜}(1). Then 𝔜̃ is a homogeneous symplectic manifold and there exists a covering {V_{i}}_{i∈I} of 𝔜 by contact charts χ_{i}: V_{i} ↪ P^{∗}Y_{i} such that, setting Ṽ_{i} = γ^{−1}V_{i}, {Ṽ_{i}}_{i∈I} is a covering of 𝔜̃ and there are commutative diagrams

\xymatrix{Ṽ_{i} \ar[d]_{γ|_{V_{i}}} \ar@{^{(}->}[r]^{_{χ_{i}}} & Ṫ^{∗}Y_{i} \ar[d]^{γ_{i}} \\ V_{i} \ar@{^{(}->}[r]^{χ_{i}} & P^{∗}Y_{i}.}

where γ_{i} is the the projection Ṫ^{∗}Y_{i} ⟶ P^{∗}Y_{i} and χ̃_{i}: Ṽ_{i} ↪ Ṫ^{∗}Y_{i} are homogeneous symplectic maps.

We denote by Mod_{loc-sys}(ℂ_{γ^{−1}Λ}) the full substack of Mod(ℂ_{γ^{−1}Λ}) consisting of local systems, and by Mod_{reg-Λ}(E_{𝔜}|_{Λ}) the full substack of Mod(E_{𝔜}|_{Λ}) consisting of modules regular along Λ

Corollary 6.4. (cf [9]) There is an equivalence of stacks

Mod_{reg-Λ}(E_{𝔜}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}) ≃ γ_{∗}Mod_{loc-sys}(ℂ_{γ^{−1}Λ}).

Proof. The sheaf of rings E_{Y} on T^{∗}Y is a subsheaf of the sheaf of rings E_{Y}^{ℝ} of [20], this last sheaf being the microlocalization along the diagonal of the sheaf O_{Y ×Y} (up to a shift and tensorizing by holomorphic forms), see [13, chapter 11] for a detailed construction. One defines the sheaf E_{Y}^{\sqrt{v},ℝ} similarly as we have defined E_{Y}^{√\overline{v}}.

With notations as in (5.2), the isomorphisms Φ_{ij} are induced by sections of a microdifferential module supported by the graph of χ_{ij}, and hence extend to isomorphisms Φ_{ij}^{ℝ}. Considering Q_{ijk} as sections of E_{Y_{i}}^{\sqrt{v},ℝ}, we get a ℂ-algebroid descent datum

Eℝ ; _{𝔜} = (\{χ̃_{i}^{−1}E_{Y_{i}}^{\sqrt{v},ℝ}|_{_{χ_{i}}(_{V_{i}})}\}_{i∈I}, \{χ̃_{i}^{−1}(Φ_{ij}^{ℝ})\}_{i,j∈I}, \{χ̃_{i}^{−1}(Q_{ijk})\}_{i,j,k∈I}),

and the associated stack Mod(Eℝ ; _{𝔜}) on 𝔜̃. The inclusion γ^{−1}E_{Y}^{√\overline{v}} ⊂ E_{Y}^{ℝ} induces a functor of extension of scalars Mod(E_{𝔜}) ⟶ γ_{∗}Mod(Eℝ ; _{𝔜}). Let us denote by M ↦ M^{ℝ} this functor.

By Theorem 6.1 there is a simple system L in Mod(E_{𝔜}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}). Consider the functor

Ψ: Mod_{reg-Λ}(E_{𝔜}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}) ⟶ γ_{∗}Mod_{loc-sys}(ℂ_{γ^{−1}Λ})

given by M ↦ Hom_{Eℝ ; _{𝔜}}(L^{ℝ}, M^{ℝ}). The functor Ψ is locally an equivalence by

the results of [11], and hence is an equivalence. q.e.d.

Example 6.5. Let 𝔜 = P^{∗}ℂ, Λ = P_{{0}}^{∗}ℂ = {pt}, γ^{−1}Λ = ℂ^{×}. In this case, simple objects of rank one are classified by ℂ^{×} ≃ ℂ/ℤ.

7 WKB-modules

The relationship between microdifferential operators on a complex contact manifold and WKB-differential operators on a complex symplectic manifold is classic, and is discussed e.g., in [1,18] in the case of cotangent bundles. This study (including the analysis of the action of quantized contact transformations) is systematically performed in [19], and here we follow their presentation.

Let X be a complex manifold, t ∈ ℂ the coordinate, and set

E_{X×ℂ,t̂} = E_{X×ℂ,∂_{t}} = {P ∈ E_{X×ℂ}; [P, ∂/∂_{t}] = 0}.

Set Ṗ^{∗}(X × ℂ) = {τ ≠ 0}, and consider the projection

ρ: Ṗ^{∗}(X × ℂ) ⟶ T^{∗}X,  (7.1)

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

W_{X} := ρ_{∗}(E_{X×ℂ,t̂}).

We similarly set W_{X}^{√\overline{v}} := ρ_{∗}(E_{X×ℂ,t̂}^{√\overline{v}}). In a local symplectic coordinate system (x; ξ) on T^{∗}X, a section P ∈ W_{X}(U) is written as a formal series

P = ∑_{j≤m} p_{j}(x; ξ)τ^{j}, p_{j} ∈ O_{T^{∗}X}(U), m ∈ ℤ,

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

One sets

k := W_{{pt}}.  (7.2)

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

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

with the condition that there exist C > 0 with |a_{j}| ≤ C^{−j}(−j)! for all j < 0.

Note that W_{X} is ℤ-filtered k-central algebra, and the principal symbol map σ_{m}: W_{X}(m) ⟶ O_{T^{∗}X} induces an isomorphism of graded algebras gr W_{X} ⟶ O_{T^{∗}X}[τ^{−1}, τ]. Note also that π^{−1}D_{X} is a subring of W_{X}.

We can now mimic Definition 4.1 for W_{X}-modules.

Definition 7.1. Let Λ be a smooth Lagrangian submanifold of T^{∗}X. Let M be a coherent W_{X}-module supported by Λ. One says that M is regular (resp. simple) along Λ if there locally exists a coherent sub-W_{X}(0)-module M_{0} of M which generates it over W_{X}, and such that M_{0}/W_{X}(−1)M_{0} is an O_{Λ}-module (resp. a locally free O_{Λ}-module of rank one).

Note that, as follows e.g. from Corollary 9.2 below, if M is regular then it is locally a finite direct sum of simple modules.

Notation 7.2. Let X be a complex manifold. We denote by O_{X}^{τ} the simple W_{X}-module along the zero-section T_{X}^{∗}X defined by O_{X}^{τ} = W_{X}/I, where I is the left ideal generated by the vector fields on X.

Proposition 7.3. (i) Any two W_{X}-modules simple along Λ are locally isomorphic. In particular, any simple module along T_{X}^{∗}X is locally isomorphic to O_{X}^{τ}.

(ii) If M, N are simple W_{X}-modules along Λ, then RHom_{W_{X}}(M, N) is a k-local system of rank one on Λ.

Proof. Since both statements are local on Λ, we may assume that X is endowed with a local coordinate system (x_{1}, …, x_{n}) and that Λ is the zerosection T_{X}^{∗}X of T^{∗}X.

(i) Any W_{X}-module simple along T_{X}^{∗}X is locally isomorphic to O_{X}^{τ}. Indeed, the proof of the theorem (due to [20]) which asserts that if Y is a complex manifold, then simple E_{Y} -modules along smooth regular involutive submanifolds of P^{∗}Y are locally isomorphic applies when replacing E_{X×ℂ} with E_{X×ℂ,t̂} (for an exposition, see [21, Ch 1, Th 6.2.1]).

(ii) By (i) we may assume that M = N = O_{X}^{τ}. The result easily follows, representing RHom_{W_{X}}(O_{X}^{τ}, O_{X}^{τ}) by the Koszul complex K^{•}(O_{X}^{τ}, (∂_{x_{1}}, …, ∂_{x_{n}})) associated with the sequence (∂_{x_{1}}, …, ∂_{x_{n}}) acting on O_{X}^{τ}. q.e.d.

Recall the projection ρ in (7.1) and note that ρ^{−1}Λ is an involutive submanifold of Ṗ^{∗}(X × ℂ).

Proposition 7.4. Let Λ be a smooth Lagrangian submanifold of T^{∗}X.

(i) Locally, there exists a Lagrangian submanifold Λ^{0} ⊂ ρ^{−1}(Λ) on which ρ induces an isomorphism Λ^{0} ≃ Λ.

(ii) Let L be a simple W_{X}-module along Λ. Then, locally there exists a simple E_{X×ℂ}-module L^{0} along Λ^{0} such that L ≃ for(L^{0}), where for denotes the forgetful functor Mod(E_{X×ℂ}) ⟶ Mod(E_{X×ℂ,t̂}).

Proof. Since the problem is local, we may assume that Λ = T_{X}^{∗}X in a system of local symplectic coordinates (x; u) ∈ T^{∗}X. In the corresponding system of homogeneous coordinates (x, t; ξ, τ) ∈ P^{∗}(X × ℂ), one has ρ^{−1}Λ = {ξ = 0}. This set is foliated by the Lagrangian submanifolds Λ^{c} = {ξ = 0, t = c}, for c ∈ ℂ. For Z ⊂ X a closed submanifold, denote by C_{Z|X} the sheaf of finite order holomorphic microfunctions on P_{Z}^{∗}X. A simple E_{X}-module along Λ^{0} is given by L^{0} = C_{X×{0}|X×ℂ}. One then immediately checks that L^{0} ≃ O_{X}^{τ}. q.e.d.

8 Contactification of symplectic manifolds

In this section, we recall some well-known facts from the specialists on contact and symplectic geometry.

Definition 8.1. A contactification of a symplectic manifold (𝔛, ω) is a complex contact manifold (𝔜, O_{𝔜}(1), α), and a morphism of complex manifolds ρ: 𝔜 ⟶ 𝔛 such that the following conditions are satisfied:

(a) the line bundle O_{𝔜}(1) has a nowhere vanishing section τ,

(b) there exists an open covering 𝔛 = ⋃_{i∈I} U_{i}, holomorphic functions t_{i} on ρ^{−1}(U_{i}) and primitives σ_{i} ∈ Ω_{U_{i}}^{1} of ω|_{U_{i}} such that χ_{i} := (ρ, t_{i}) gives an isomorphism χ_{i}: ρ^{−1}(U_{i}) \underrightarrow{∼} U_{i} × ℂ, and dt_{i} + ρ^{∗}(σ_{i}) = (α/τ)|_{ρ^{−1}(U_{i})}.

Lemma 8.2. A contactification ρ: 𝔜 ⟶ 𝔛 has a structure of ℂ-principal bundle.

Proof. With notations as in Definition 8.1, set χ_{ij} = χ_{j}χ_{i}^{−1}|_{U_{ij}}: U_{ij} × ℂ ⟶ U_{ij} × ℂ. We have χ_{ij} = (id_{U_{ij}}, θ_{ij}) for some transition functions θ_{ij}: U_{ij} × ℂ ⟶ ℂ. Since d(t_{i} − t_{j}) = ρ^{∗}(σ_{i} − σ_{j}), up to shrinking the covering we may assume that σ_{i} − σ_{j} = df_{ij} for some functions f_{ij} ∈ O_{U_{ij}}. The transition functions are the translations θ_{ij}(p, t) = t + f_{ij}(p) + c_{ij} for some c_{ij} ∈ ℂ. q.e.d.

Lemma 8.3. A symplectic manifold (𝔛, ω) admits a contactification if and only if the de Rham cohomology class [ω] ∈ H^{2}(𝔛; ℂ_{𝔛}) vanishes.

Proof. (a) Let 𝔛 = ⋃_{i∈I} U_{i} be a covering such that ω|_{U_{i}} has a primitive σ_{i} ∈ Ω_{U_{i}}^{1}. Up to shrinking the covering we may assume that σ_{i} − σ_{j} = df_{ij} for some functions f_{ij} ∈ O_{U_{ij}}. Then d(f_{ij} + f_{jk} − f_{ik}) = 0, and ℂ_{U_{ijk}} ∋ c_{ijk} = f_{ij} + f_{jk} − f_{ik} is a Cech cocycle representing [ω] ∈ H^{2}(𝔛; ℂ_{𝔛}). Since [ω] = 0, we have c_{ijk} = c_{ij} + c_{jk} − c_{ik}. Setting f̃_{ij} = f_{ij} − c_{ij}, we have f̃_{ij} + f̃_{jk} − f̃_{ik} = 0. Endow U_{i} × ℂ with the contact form dt + σ_{i}. A contactification of 𝔛 is thus given by the ℂ-principal bundle 𝔜 ⟶ 𝔛 with local charts U_{i}×ℂ, and transition functions over U_{ij} given by (p, t) ↦ (p, t + f̃_{ij}(p)).

(b) Let ρ: 𝔜 ⟶ 𝔛 be a contactification, and use notations as in Definition 8.1. We have d(α/τ) = ρ^{∗}ω and hence [ρ^{∗}ω] = 0 in H^{2}(𝔜; ℂ_{𝔜}). Then [ω] = 0 in H^{2}(𝔛; ℂ_{𝔛}) \underrightarrow{∼} H^{2}(𝔜; ℂ_{𝔜}). q.e.d.

Lemma 8.4. Let (𝔛, ω) be a symplectic manifold and Λ a Lagrangian submanifold. After replacing 𝔛 with a neighborhood of Λ there exists a contactification ρ: 𝔜 ⟶ 𝔛 and a Lagrangian submanifold Λ^{0} ⊂ 𝔜 on which ρ induces an isomorphism Λ^{0} ≃ Λ.

Proof. The restriction map H^{2}(𝔛, ℂ_{𝔛}) ⟶ H^{2}(Λ; ℂ_{Λ}) is given by [ω] ↦ [j^{∗}ω], where j: Λ ↪ 𝔛 is the embedding. Since ω|_{Λ} = 0, there exists an open neighborhood U ⊃ Λ such that [ω|_{U}] = 0. Thus, up to replacing 𝔛 with U, we can assume that [ω] = 0.

Let us adapt the arguments in part (a) of the proof of Lemma 8.3 above. Let 𝔛 = ⋃_{i∈I} U_{i} be a covering such that ω|_{U_{i}} has a primitive σ_{i} ∈ Ω_{U_{i}}^{1}. We may assume that σ_{i}|_{Λ} = 0. Let f_{ij} ∈ O_{U_{ij}} satisfy σ_{i} − σ_{j} = df_{ij}. We may assume moreover that f_{ij}|_{Λ} = 0. Since f_{ij} + f_{jk} − f_{ik} is locally constant on Λ, it must vanish on Λ. Thus, the ℂ-principal bundle ρ: 𝔜 ⟶ 𝔛 is described by the local charts U_{i} × ℂ, with transition functions (p, t) ↦ (p, t + f_{ij}(p)) and contact form dt + σ_{i}. We define Λ^{0} by Λ^{0}|_{U_{i}×ℂ} = Λ|_{U_{i}} × {0}. q.e.d.

9 Simple holonomic modules on symplectic manifolds

Let us start by recalling the construction of [19] of the stack of WKB-modules on a symplectic manifold, in the special case where there exists a contactification.

Let ρ: 𝔜 ⟶ 𝔛 be a contactification of a complex symplectic manifold. Denote by

E_{𝔜,τ} = ({A_{i}}_{i∈I}, {f_{ij}}_{i,j∈I}, {a_{ijk}}_{i,j,k∈I})

the ℂ-algebroid descent datum on 𝔜 given by Proposition 5.5. Consider the ℂ-algebroid descent datum on 𝔛

W_{𝔛} = ({ρ_{∗}A_{i}}_{i∈I}, {ρ_{∗}f_{ij}}_{i,j∈I}, {ρ_{∗}a_{ijk}}_{i,j,k∈I}).

The stack Mod(W_{𝔛}) associated with this ℂ-algebroid descent datum is the stack of WKB-modules in [19].

One defines the notion of good W_{𝔛}-module similarly as for E_{𝔜}-modules.

Here, we prove the existence of twisted simple holonomic modules along smooth Lagrangian submanifolds of complex symplectic manifolds.

Theorem 9.1. Let 𝔛 be a complex symplectic manifold and let Λ ⊂ 𝔛 be a Lagrangian submanifold. There exists a module L ∈ Mod(W_{𝔛}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}) which is simple along Λ. Moreover, L is good.

Proof. By Lemma 8.3 there exist a contactification ρ: 𝔜 ⟶ 𝔛 and a Lagrangian submanifold Λ^{0} ⊂ 𝔜 on which ρ induces an isomorphism Λ^{0} ≃ Λ. By Theorem 6.1 there exists L^{0} ∈ Mod(E_{𝔜}|_{Λ^{0}} ⊗_{ℂ}ℂ\sqrt{Ω_{Λ^{0}}}) which is simple along

Λ^{0}. As in Proposition 7.4 we then set L = for(L^{0}), where

for : Mod(E_{𝔜}|_{Λ^{0}} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ^{0}}}) ⟶ Mod(E_{𝔜,t̂}|_{Λ^{0}} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ^{0}}})
≃ Mod(W_{𝔛}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}})

is the natural forgetful functor. q.e.d.

Corollary 9.2. There is a k-equivalence of stacks

Mod_{reg-Λ}(W_{𝔛}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}) ≃ Mod_{loc-sys}(k_{Λ}).  (9.1)

Proof. By Theorem 9.1, there exists a simple module L in Mod(W_{𝔛}|_{Λ} ⊗_{ℂ} ℂ\sqrt{Ω_{Λ}}). By Proposition 7.3, a functor (9.1) is given by M ↦ Hom_{W_{𝔛}}(L, M). Proving that it is an equivalence is a local problem, and so we may assume 𝔛 = T^{∗}X, Λ = T_{X}^{∗}X. Then a quasi inverse is given by F ↦ F ⊗_{k} O_{X}^{τ}. q.e.d.

Remark 9.3. Note that in the real setting, a link between simple holonomic modules on real Lagrangian submanifolds and Fourier distributions, Maslov index, etc. is investigated in [17].

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