Radon-Penrose transform for D-modules

Andrea D’Agnolo  Pierre Schapira

September 6, 1994

Institut de Mathématiques Analyse Algébrique Université Pierre et Marie Curie Case 247 4, place Jussieu F-75252 Paris Cedex 05

email: dagnolo@ccr.jussieu.fr, schapira@ccr.jussieu.fr

• Proposed running head: Radon-Penrose transform for D-modules

• Proofs should be sent to: Andrea D’Agnolo Institut de Mathématiques Analyse Algébrique Université Pierre et Marie Curie Case 247 4, place Jussieu F-75252 Paris Cedex 05

Abstract

Let X ⟵_{f} S ⟶_{g} Y be a correspondence of complex analytic manifolds, F be a sheaf on X, and M be a coherent D_{X}-module. Consider the associated sheaf theoretical and D-module integral transforms given by Φ_{S}F = Rg_{!}f^{−1}F [d] and \underline{Φ}_{S}M = \underline{g}_{!}\underline{f}^{−1}M, where Rg_{!} and f^{−1} (resp. \underline{g}_{!} and \underline{f}^{−1}) denote the direct and inverse image functors for sheaves (resp. for D-modules), and d = d_{S} − d_{Y} is the difference of dimension between S and Y. In this paper, assuming that f is smooth, g is proper, and (f, g) is a closed embedding, we prove some general adjunction formulas for the functors Φ_{S} and \underline{Φ}_{S}. Moreover, under an additional geometrical hypothesis, we show that the transformation \underline{Φ}_{S} establishes an equivalence of categories between coherent D_{X}-modules, modulo flat connections, and coherent D_{Y} -modules with regular singularities along an involutive manifold V, modulo flat connections (here V is determined by the geometry of the correspondence). Applications are given to the case of Penrose’s twistor correspondence.

AMS classification: 32L25, 32C38, 58G05

Contents

1 Introduction 5

2 Adjunction formulas 7 2.1 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 D-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Vanishing theorems 16 3.1 E-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Vanishing theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 A regularity theorem and an equivalence of categories 21 4.1 Modules with regular singularities . . . . . . . . . . . . . . . . . . . . 21 4.2 A regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 An equivalence of categories . . . . . . . . . . . . . . . . . . . . . . . 26

5 Applications 31 5.1 The twistor correspondence (holomorphic solutions) . . . . . . . . . . 31 5.2 The twistor correspondence (hyperfunction solutions) . . . . . . . . . 33

1 Introduction

The Penrose correspondence is an integral transformation which interchanges global sections of line bundles on some flag manifolds, with holomorphic solutions of partial differential equations on other flag manifolds (see [6], [1]). For example, consider the twistor correspondence:

ℙ  𝕄,^{𝔽 N N N Q_{f}  _{ȷȷȷ fi}_{g}}

(1.1)

where 𝔽 = 𝔽_{1,2}(𝕋) is the flag manifold of type (1, 2) associated to a four-dimensional complex vector space 𝕋, ℙ = 𝔽_{1}(𝕋) is a projective three-space, and 𝕄 = 𝔽_{2}(𝕋). The projections are given by f(L_{1}, L_{2}) = L_{1}, g(L_{1}, L_{2}) = L_{2}, where L_{1} ⊂ L_{2} ⊂ 𝕋 are complex subspaces of dimension one and two respectively, defining an element (L_{1}, L_{2}) of 𝔽. Since 𝕄 is identified with the four-dimensional compactified complexified Minkowski space, the family of massless field equations on the Minkowski space gives rise to a family of differential operators acting between sections of holomorphic bundles on 𝕄. This is a family, denoted here by _{h}, which is parameterized by a halfinteger h called helicity, and which includes Maxwell’s wave equation, Dirac-Weyl neutrino equations and Einstein linearized vacuum equations.

The Penrose transform associated to the correspondence (1.1) allows to represent the holomorphic solutions of the equation _{h}ϕ = 0 on some open subsets U ⊂ 𝕄 in terms of cohomology classes of line bundles on Û = f(g^{−1}(U)) ⊂ ℙ. More precisely, recall that the line bundles on ℙ are given, for k ∈ ℤ, by the −k-th tensor powers O_{ℙ}(k) of the tautological bundle. Set h(k) = −(1 + k/2), and for x ∈ ℙ, set x̂ = g(f^{−1}(x)). We then have the result of Eastwood, Penrose and Wells [6] below.

Theorem 1.1 Let U ⊂ 𝕄 be an open subset such that:

(1.2) U ∩ x̂ is connected and simply connected for every x ∈ Û.

Then, for k < 0, the natural morphism associated to (1.1), which maps a one-form on Û to the integral along the fibers of g of its inverse image by f, induces an isomorphism:

H^{1}(Û; O_{ℙ}(k)) ⟶^{∼} ker(U; _{h(k)}).

In this paper we shall formulate the Penrose correspondence in the language of sheaves and D-modules. First of all, we can rephrase the above construction in a more general setting as follows. Consider a correspondence:

S_{N N N Q}_{f}  _{ȷȷȷ fi}_{g} ; X  Y,

where all manifolds are complex analytic, f is smooth, g is proper, and where (f, g) induces a closed embedding S ↪ X × Y. Set d_{S} = dim^{ℂ} S, d_{S/Y} = d_{S} − d_{Y}.

Let us define the transform of a sheaf F on X (more generally, of an object of the derived category of sheaves) as Φ_{S}F = Rg_{!}f^{−1}F [d_{S/Y}], and define the transform of a coherent D_{X}-module M as \underline{Φ}_{S}M = \underline{g}_{∗}\underline{f}^{−1}M, where \underline{g}_{∗} and \underline{f}^{−1} denote the direct and inverse images in the sense of D-module theory. We also consider Φ_{S}G = Rf_{!}g^{−1}G[d_{S/X}], for a sheaf G on Y. One then proves the formula:

Φ_{S}RHom_{D_{X}}(M, O_{X}) ≃ RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y}), (1.3)

from which one deduces the following formula, where G denotes a sheaf on Y:

RΓ(X; RHom_{D_{X}} (M ⊗ Φ_{S}G, O_{X}))[d_{X}] ≃ RΓ(Y; RHom_{D_{Y}} (\underline{Φ}_{S}M ⊗ G, O_{Y}))[d_{Y}]. (1.4)

Let F be a holomorphic vector bundle on X, denote by F^{∗} its dual, and set DF^{∗} = D_{X} ⊗_{O_{X}} F^{∗}. When applying (1.4) to the case of M = DF^{∗} and G = ℂ_{U}, the constant sheaf on an open subset U ⊂ Y satisfying suitable hypotheses, one gets the formula:

RΓ(Û; F) ≃ RΓ(U; RHom_{D_{Y}} (\underline{Φ}_{S}DF^{∗}, O_{Y}))[−d_{S/Y}]. (1.5)

In other words, the cohomology of F on Û is isomorphic to the holomorphic solutions on U of some complex of coherent D_{Y} -modules, namely the complex \underline{Φ}_{S}DF^{∗}.

In the particular case of the twistor correspondence, the above results show that Theorem 1.1 is better understood by saying that the D-module transform of DO_{ℙ}(−k) (for k < 0) is the D_{𝕄}-module associated to the differential operator _{h(k)}. Moreover, formula (1.4) shows that each of the many problems encountered in literature can be split into two different ones:

(i) to calculate the sheaf theoretical transform Φ_{S}G of G,

(ii) to calculate the D-module transform \underline{Φ}_{S}M of M.

The calculation of Φ_{S}G relies on the particular geometry considered (see section 5.2 for an example, where we easily recover Wells’s result on hyperfunction solutions).

The calculation of \underline{Φ}_{S}M leads to more difficult problems. For instance, notice that in general \underline{Φ}_{S}M is a complex, not necessarily concentrated in degree zero. This implies many technical difficulties when interpreting the cohomology groups of the right hand side of (1.4). In this paper we give several properties of \underline{Φ}_{S}M, which hold under geometrical hypotheses that we will formulate later:

(i) H^{0}(\underline{Φ}_{S}M) is a coherent D_{Y} -module with regular singularities along an involutive manifold V of the cotangent bundle T^{∗}Y given by the geometry,

(ii) for j ≠ 0, H^{j}(\underline{Φ}_{S}M) is a locally free O_{Y} -module of finite rank endowed with a flat connection,

(iii) in the case M = DF^{∗}, for a complex vector bundle F, we give several formulas similar to (1.5), and in particular we prove the germ formula (where ŷ = f(g^{−1}(y))):

RΓ(ŷ; F) ≃ RHom_{D_{Y}} (\underline{Φ}_{S}DF^{∗}, O_{Y})_{y}[−d_{S/Y}],

from which we deduce that H^{j}(\underline{Φ}_{S}DF^{∗}) = 0 for j ≠ 0 if and only if (Y being connected) there exists y ∈ Y such that H^{j}(ŷ; F) = 0 for j < d_{S/Y}.

Then, and it is our main result, we prove that (under suitable hypotheses which are satisfied in the twistor case) the transform \underline{Φ}_{S} induces an equivalence of categories between coherent D-modules on X modulo flat connections, and coherent D-modules on Y with regular singularities along the involutive submanifold V of (i), modulo flat connections. When applied to the twistor case, our results show in particular that any D-module on the Minkowski space with regular singularities along the characteristic variety of the wave equation, may be obtained (up to flat connections) as the image of a coherent D-module on ℙ.

The results of this paper were announced in [3]. When writing this paper we benefitted from many classical works on the Penrose correspondence. In particular, let us mention the books [19], [1], [27] and the papers [6], [5], [28], [29]. Note that a microlocal approach in the study of correspondences was initiated in the paper [7] of Guillemin and Sternberg.

Finally, we would like to thank Jean-Pierre Schneiders for fruitful discussions.

2 Adjunction formulas

2.1 Sheaves

Let X be a real analytic manifold. We denote by D(X) the derived category of the category of complexes of sheaves of ℂ-vector spaces on X, by D^{b}(X) the full triangulated subcategory of D(X) whose objects have bounded cohomology, and we refer to [17] for a detailed exposition on sheaves, in the framework of derived categories.

If A ⊂ X is a locally closed subset, we denote by ℂ_{A} the sheaf on X which is the constant sheaf on A with stalk ℂ, and zero on X \ A. We will consider the classical six operations in the derived category of sheaves of ℂ-vector spaces f^{−1}, Rf_{!}, ⊗, Rf_{∗}, f^{!}, RHom. We denote by ω_{Y/X} the relative dualizing complex ω_{Y/X} = f^{!}ℂ_{X}. Recall that ω_{Y/X} ≃ or_{Y/X}[d], where or_{Y/X} is the relative orientation sheaf on Y, and d = dim^{ℝ} Y − dim^{ℝ} X, dim^{ℝ} X denoting the dimension of X. We use the notations D′_{X}(·) = RHom(·, ℂ_{X}) and D_{X}(·) = RHom(·, ω_{X}), where ω_{X} = ω_{X/{pt}}. We denote by a_{X} : X ⟶ {pt} the map from X to the set consisting of a single element.

In the rest of this section, all manifolds and morphisms of manifolds will be complex analytic. We denote by d_{S} the complex dimension of a manifold S. Given a morphism f : S ⟶ X of complex manifolds, we set for short d_{S/X} = d_{S} − d_{X}.

Consider a correspondence of complex analytic manifolds:

S_{N N N Q}_{f}  _{ȷȷȷ fi}_{g} ; X  Y.

(2.1)

Definition 2.1 For F ∈ D^{b}(X), we set:

Φ_{S}F = Rg_{!}f^{−1}(F)[d_{S/Y}], Ψ_{S}F = Rg_{∗}f^{!}(F)[−d_{S/X}].

For G ∈ D^{b}(Y), we similarly define:

Φ_{S}G = Rf_{!}g^{−1}(G)[d_{S/X}], Ψ_{S}G = Rf_{∗}g^{!}(G)[−d_{S/Y}].

In other words, we denote by

S̃ N N N Q_{g}  _{ȷȷȷ fi}_{f}

Y  X

(2.2)

the correspondence deduced from (2.1) by interchanging X and Y.

Lemma 2.2 Let F ∈ D^{b}(X) and G ∈ D^{b}(Y). Then we have the isomorphisms:

(2.3)  Ra_{X}_{∗}RHom(Φ_{S}G, F) ≃ Ra_{Y} _{∗}RHom(G, Ψ_{S}F),
(2.4)  Ra_{X}_{!}(Φ_{S}G ⊗ F)[d_{X}] ≃ Ra_{Y} _{!}(G ⊗ Φ_{S}F)[d_{Y}],
(2.5)  Ψ_{S}(D_{X}F)[d_{Y}] ≃ D_{Y} (Φ_{S}F)[d_{X}].

Proof: All the above isomorphisms are easy consequences of classical adjunction formulas, such as the Poincaré-Verdier duality formula (see e.g. [17, chapters II and III]). For example, in order to prove (2.4) one considers the sequence of isomorphisms:

Ra_{X}_{!}(Φ_{S}G ⊗ F)[d_{X}] = Ra_{X}_{!}(Rf_{!}g^{−1}G ⊗ F)[d_{S}]
≃ Ra_{X}_{!}Rf_{!}(g^{−1}G ⊗ f^{−1}F)[d_{S}]
≃ Ra_{Y} _{!}Rg_{!}(g^{−1}G ⊗ f^{−1}F)[d_{S}]
≃ Ra_{Y} _{!}(G ⊗ Rg_{!}f^{−1}F)[d_{S}]
= Ra_{Y} _{!}(G ⊗ Φ_{S}F)[d_{Y}].
□

2.2 D-modules

Let O_{X} denote the sheaf of holomorphic functions on a complex manifold X, Ω_{X} the sheaf of holomorphic forms of maximal degree, and D_{X} the sheaf of rings of holomorphic linear differential operators. We refer to [15], [23] for the theory of D-modules (see [24] for a detailed exposition).

Denote by Mod(D_{X}) the category of left D_{X}-modules, and by Mod_{coh}(D_{X}) the thick abelian subcategory of coherent D_{X}-modules. Following [25], we say that a coherent D_{X}-module M is good if, in a neighborhood of any compact subset of X, M admits a finite filtration by coherent D_{X}-submodules M_{k} (k = 1, …, l) such that each quotient M_{k}/M_{k−1} can be endowed with a good filtration. We denote by Mod_{good}(D_{X}) the full subcategory of Mod_{coh}(D_{X}) consisting of good D_{X}- modules. This definition ensures that Mod_{good}(D_{X}) is the smallest thick subcategory of Mod(D_{X}) containing the modules which can be endowed with good filtrations on a neighborhood of any compact subset of X. Note that in the algebraic case, coherent D-modules are good.

Denote by D^{b}(D_{X}) the derived category of the category of bounded complexes of left D_{X}-modules, and by D_{coh}^{b}(D_{X}) (resp. by D_{good}^{b}(D_{X})) its full triangulated subcategory whose objects have cohomology groups belonging to Mod_{coh}(D_{X}) (resp. to Mod_{good}(D_{X})).

Let f : Y ⟶ X be a morphism of complex manifolds. We denote by \underline{f}^{−1} and \underline{f}_{∗} the inverse and direct images in the sense of D-modules. Hence, for M ∈ D^{b}(D_{X}) and N ∈ D^{b}(D_{Y}):

\underline{f}^{−1}M = D_{Y →X} ⊗_{f^{−1}D_{X}}^{L} f^{−1}M, \underline{f}_{∗}N = Rf_{∗}(D_{X←Y} ⊗_{D_{Y}}^{L} N),

where D_{Y →X} and D_{X←Y} are the transfer bimodules. We denote by \underline{×} the exterior tensor product, and we also use the notation:

\underline{D}_{X}M = RHom_{D_{X}}(M, K_{X}),

where K_{X} is the dualizing complex for left D_{X}-modules, K_{X} = D_{X} ⊗_{O_{X}} Ω_{X}^{⊗−1}[d_{X}].

Proposition 2.3 Let M ∈ D_{good}^{b}(D_{X}), N ∈ D^{b}(D_{Y}), and G ∈ D^{b}(Y). Assume that f is non-characteristic for M. Then \underline{f}^{−1}M is good, and we have the isomorphisms:

(2.6)  \underline{D}\underline{f}^{−1}M ≃ \underline{f}^{−1}\underline{D}M,
(2.7) Rf_{∗}RHom_{D_{Y}} (\underline{f}^{−1}M, N [d_{Y/X}]) ≃ RHom_{D_{X}}(M, \underline{f}_{∗}N),
(2.8)  f^{−1}RHom_{D_{X}}(M, O_{X}) ≃ RHom_{D_{Y}} (\underline{f}^{−1}M, O_{Y}),
(2.9)  RHom_{D_{X}}(M, Rf_{!}G ⊗ O_{X}) ≃ Rf_{!}RHom_{D_{Y}} (\underline{f}^{−1}M, G ⊗ O_{Y}),

RHom_{D_{X}}(M, RHom(Rf_{!}G, O_{X})) (2.10)

≃ Rf_{∗}RHom_{D_{Y}} (\underline{f}^{−1}M, RHom(G, O_{Y}))[2d_{Y/X}].

Proof: The fact that \underline{f}^{−1}M is good and the first isomorphism are results of [23]. The second isomorphism is easily deduced from the first one, and the third isomorphism is the Cauchy-Kowalevski-Kashiwara theorem. Let us prove (2.9). Let us set for short Sol(M) = RHom_{D_{X}}(M, O_{X}). Then we have the chain of isomorphisms:

RHom_{D_{X}}(M, Rf_{!}G ⊗ O_{X}) ≃ Rf_{!}G ⊗ Sol(M)
≃ Rf_{!}(G ⊗ f^{−1}Sol(M))
≃ Rf_{!}(G ⊗ Sol(\underline{f}^{−1}M)),

where the last isomorphism follows from (2.8). To prove (2.10), consider the chain of isomorphisms:

RHom_{D_{X}}(M, RHom(Rf_{!}G, O_{X})) ≃ RHom(Rf_{!}G, Sol(M))
≃ Rf_{∗}RHom(G, f^{!}Sol(M))
≃ Rf_{∗}RHom(G, f^{−1}Sol(M))[2d_{Y/X}]
≃ Rf_{∗}RHom(G, Sol(\underline{f}^{−1}M))[2d_{Y/X}],

where, in order to prove the third isomorphism, we have used Proposition 5.4.13 and Theorem 11.3.3 of [17]. □

Proposition 2.4 Let N ∈ D_{good}^{b}(D_{Y}), M ∈ D^{b}(D_{X}), and F ∈ D^{b}(X). Assume f is proper on supp N. Then \underline{f}_{∗}N is good, and:

(2.11)  \underline{D}\underline{f}_{∗}N ≃ \underline{f}_{∗}\underline{D}N ,
(2.12)  Rf_{∗}RHom_{D_{Y}} (N , \underline{f}^{−1}M[d_{Y/X}]) ≃ RHom_{D_{X}}(\underline{f}_{∗}N , M),
(2.13)  Rf_{∗}RHom_{D_{Y}} (N , O_{Y})[d_{Y/X}] ≃ RHom_{D_{X}}(\underline{f}_{∗}N , O_{X}),
(2.14) Rf_{!}RHom_{D_{Y}} (N , f^{−1}F ⊗ O_{Y})[d_{Y/X}] ≃ RHom_{D_{X}}(\underline{f}_{∗}N , F ⊗ O_{X}),

Rf_{∗}RHom_{D_{Y}} (N , RHom(f^{−1}F, O_{Y}))[d_{Y/X}] (2.15)

≃ RHom_{D_{X}}(\underline{f}_{∗}N , RHom(F, O_{X})).

Proof: The fact that \underline{f}_{∗}N is good, and the first isomorphism are results of [14], [12], [26], and [25]. The second and third isomorphisms follow from the first one. To prove (2.14), consider the chain of isomorphisms:

Rf_{!}(f^{−1}F ⊗ Sol(N))[d_{Y/X}] ≃ F ⊗ Rf_{!}Sol(N)[d_{Y/X}]
≃ F ⊗ Sol(\underline{f}_{∗}N).

To prove (2.14), consider the chain of isomorphisms:

Rf_{∗}RHom(f^{−1}F, Sol(N))[d_{Y/X}] ≃ RHom(F, Rf_{∗}Sol(N))[d_{Y/X}]
≃ RHom(F, Sol(\underline{f}_{∗}N)).
□

2.3 Correspondences

Instead of considering morphisms, we shall now consider correspondences of complex analytic manifolds:

S_{N N N Q}_{f}  _{ȷȷȷ fi}_{g} ; X  Y.

(2.16)

Definition 2.5 For M ∈ D^{b}(D_{X}), we set:

\underline{Φ}_{S}M = \underline{g}_{∗}\underline{f}^{−1}M, \underline{Ψ}_{S}M = \underline{Φ}_{S}M[d_{Y} − d_{X}].

For N ∈ D^{b}(D_{Y}) we similarly define:

\underline{Φ}_{S}N = \underline{f}_{∗}\underline{g}^{−1}N , \underline{Ψ}_{S}N = \underline{Φ}_{S}N [d_{X} − d_{Y}].

As an immediate consequence of Propositions 2.3 and 2.4, we get

Proposition 2.6 Let M ∈ D_{good}^{b}(D_{X}), N ∈ D^{b}(D_{Y}), and G ∈ D^{b}(Y). Assume that f is non-characteristic for M, and that g is proper on f^{−1} supp M. Then \underline{Φ}_{S}M ∈ D_{good}^{b}(D_{Y}), and:

(2.17)  \underline{Φ}_{S}(\underline{D}_{X}M) ≃ \underline{D}_{Y} (\underline{Φ}_{S}M),
(2.18)  Ra_{X}_{∗}RHom_{D_{X}}(M, \underline{Ψ}_{S}N) ≃ Ra_{Y} _{∗}RHom_{D_{Y}} (\underline{Φ}_{S}M, N),
(2.19)  Φ_{S}RHom_{D_{X}}(M, O_{X}) ≃ RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y}),

Ra_{X}_{!}RHom_{D_{X}}(M, Φ_{S}G ⊗ O_{X})[d_{X}] (2.20)

≃ Ra_{Y} _{!}RHom_{D_{Y}} (\underline{Φ}_{S}M, G ⊗ O_{Y})[d_{Y}],

Ra_{X}_{∗}RHom_{D_{X}}(M ⊗ Φ_{S}G, O_{X})[d_{X}] (2.21)

≃ Ra_{Y} _{∗}RHom_{D_{Y}} (\underline{Φ}_{S}M ⊗ G, O_{Y})[d_{Y}].

As already mentioned in the introduction, this result allows us to distinguish between two kind of problems arising in the Penrose transform:

(i) to compute the sheaf theoretical transform of G,

(ii) to compute the D-module transform of M.

The first problem is of a topological nature, and under reasonable hypotheses is not very difficult (a first example appears in Corollary 2.9 below). The study of \underline{Φ}_{S}M is, in general, a more difficult problem. For instance, \underline{Φ}_{S}M is a complex of D-modules, and is not necessarily concentrated in degree zero. This does not affect the formulas as long as we use derived categories, but things may become rather complicated when computing explicitly cohomology groups.

In the next sections we will study the transform \underline{Φ}_{S}. We begin here with some easy corollaries of Proposition 2.6.

For x ∈ X, y ∈ Y, A ⊂ X and B ⊂ Y, we set for short:

\begin{aligned}x̂ & = & g(f^{−1}(x)), & Â = g(f^{−1}(A)), \\ ŷ & = & f(g^{−1}(y)), & B̂ = f(g^{−1}(B)). \end{aligned}

Definition 2.7 (i) We say that a topological space A is globally cohomologically trivial (g.c.t. for short) if the natural morphism:

ℂ ⟶ RΓ(A; ℂ_{A})

is an isomorphism.

(ii) We say that a locally closed subset A ⊂ X is S-trivial if A ∩ ŷ is g.c.t. for every y ∈ Â.

Notice that contractible spaces are g.c.t. Moreover, a C^{0}-manifold A is g.c.t. if and only if the natural morphism

RΓ_{c}(A; ω_{A}) ⟶ ℂ (2.22)

is an isomorphism (see [17, Remark 3.3.10]).

Recall that one says that a morphism f : S ⟶ X of real analytic manifolds is smooth at s ∈ S if the tangent map f′(s) is surjective, that f is an immersion if f′(s) is injective, and that f is an embedding if it is both injective and an immersion.

In the following, we will make some of the hypotheses:

(2.23) f is smooth and g is proper, (2.24) (f, g) : S ⟶ X × Y is a closed embedding.

Lemma 2.8 Assume (2.23) and (2.24).

(i) Let U ⊂ Y be a S̃-trivial open subset. Then, there is a natural isomorphism:

Φ_{S}(ℂ_{U}) ≃ ℂ_{U}[−d_{S/X}].

(ii) Let K ⊂ Y be a S̃-trivial compact subset. Then, there is a natural isomor-

phism:

Φ_{S}(ℂ_{K}) ≃ ℂ_{K}[d_{S/X}]

Proof: (i) One has Φ_{S}(ℂ_{U}) ≃ Rf_{!}ℂ_{g^{−1}(U)}[d_{S/X}]. Setting f_{U} = f|_{g^{−1}(U)}, it remains to check that Rf_{U}_{!}ℂ_{g^{−1}(U)} ≃ ℂ_{U}[−2d_{S/X}]. Since f_{U} is smooth, one has

Rf_{U}_{!}ℂ_{g^{−1}(U)} ≃ Rf_{U}_{!}f_{U}^{−1}ℂ_{U}
≃ Rf_{U}_{!}f_{U}^{!} ℂ_{U}[−2d_{S/X}].

By hypothesis (2.24), g induces an isomorphism from g^{−1}(U) ∩ f^{−1}(x) to U ∩ x̂. Hence, S̃-triviality of U implies that the fibers of f_{U} are g.c.t. Then, the natural morphism

Rf_{U}_{!}f_{U}^{!} ℂ_{U} ⟶ ℂ_{U}

is an isomorphism by (2.22).

(ii) One has Φ_{S}(ℂ_{K}) ≃ Rf_{!}ℂ_{g^{−1}(K)}[d_{S/X}]. Setting f_{K} = f|_{g^{−1}(K)}, it remains to check that Rf_{K}_{!}ℂ_{g^{−1}(K)} ≃ ℂ_{K}. One has

Rf_{K}_{!}ℂ_{g^{−1}(K)} ≃ Rf_{K}_{∗}f_{K}^{−1}ℂ_{K}.

Moreover, the natural morphism ℂ_{K} ⟶ Rf_{K}_{∗}f_{K}^{−1}ℂ_{K} is an isomorphism by the hypothesis that K is S̃-trivial. □

By Proposition 2.6 and Lemma 2.8, we get the corollary below.

Corollary 2.9 Assume (2.23) and (2.24). Let M ∈ D_{good}^{b}(D_{X}).

(i) Let U ⊂ Y be a S̃-trivial open subset. Then, there are natural isomorphisms:

RΓ_{c}(Û; RHom_{D_{X}} (M, O_{X})) ≃ RΓ_{c}(U; RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y}))[−d_{S/Y} + 2d_{S/X}],
RΓ(Û; RHom_{D_{X}} (M, O_{X})) ≃ RΓ(U; RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y}))[−d_{S/Y}].

(ii) Let K ⊂ Y be a S̃-trivial compact subset. Then, there are natural isomorphisms:

RΓ(K̂; RHom_{D_{X}} (M, O_{X})) ≃ RΓ(K; RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y}))[−d_{S/Y}],
RΓ_{K}(X; RHom_{D_{X}} (M, O_{X})) ≃ RΓ_{K}(Y; RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y}))[2d_{S/X} − d_{S/Y}].

(iii) For y ∈ Y we have the germ formula:

RΓ(ŷ; RHom_{D_{X}}(M, O_{X})) ≃ RHom_{D_{Y}} (\underline{Φ}_{S}M, O_{Y})_{y}[−d_{S/Y}].

Notation 2.10 Let F be a locally free O_{X}-module of finite rank (a complex vector bundle). We set:

F^{∗} = Hom_{O_{X}}(F, O_{X}),
DF^{∗} = D_{X} ⊗_{O_{X}} F^{∗}.

Notice that DF^{∗} is a locally free D_{X}-module whose holomorphic solutions are given by:

RHom_{D_{X}}(DF^{∗}, O_{X}) ≃ F,

where the last isomorphism is ℂ-linear, but not O-linear.

Replacing M by DF^{∗} in Corollary 2.9, and with the notations and hypotheses of this corollary, we get in particular the isomorphisms:

(2.25)  RΓ(Û; F) ≃ RΓ(U; RHom_{D_{Y}} (\underline{Φ}_{S}(DF^{∗}), O_{Y}))[−d_{S/Y}],
(2.26)  RΓ(ŷ; F) ≃ RHom_{D_{Y}} (\underline{Φ}_{S}(DF^{∗}), O_{Y})_{y}[−d_{S/Y}].

Remark 2.11 There are many other interesting applications of the results in this section that we will not develop here. Among others, let us mention the following:

(i) Several authors have considered the formal completion of a locally free sheaf F along ŷ, or more generally along a complex submanifold of X (see [21], [13], [8], [9]). One could also incorporate this point of view in our formalism, and, for example, prove the formula below, similar to (2.26):

RΓ(ŷ; Fˆ|ŷ) ≃ RHom_{D_{Y}} (\underline{Φ}_{S}(DF^{∗}), Ô_{Y,y})[−d_{S/Y}],

where Fˆ|ŷ is the formal completion of F along the submanifold ŷ of X, and Ô_{Y,y} the formal completion of O_{Y} at y.

(ii) In (2.17) we have given a duality formula that we shall not develop here. We refer to [6], in which the duality is used to treat the positive helicity case as potentials modulo gauges.

(iii) The results of [10] concerning the group SO(8) could be reformulated in our language.

(iv) In our paper [4], we treat along these lines the classical projective duality, in which Y = ℙ^{n}, X = (ℙ^{n})^{∗} and S ⊂ X × Y is given by the incidence relation S = {(x, ξ) ∈ ℙ^{n} × (ℙ^{n})^{∗}; ⟨x, ξ⟩ = 0}.

2.4 Kernels

In this section, we shall describe an equivalent construction of the transforms Φ_{S}, \underline{Φ}_{S} and their generalizations.

On a product space X × Y, we denote by q_{1} and q_{2} the projections on X and Y respectively, and by r : X × Y ⟶ Y × X the map r(x, y) = (y, x).

Consider the correspondence (2.16). Assuming (2.24), we identify S to a closed submanifold of X × Y, and we set S̃ = r(S) ⊂ Y × X. We denote by B_{S|X×Y} the holonomic D_{X×Y} -module associated to S:

B_{S|X×Y} = H_{[S]}^{d_{X}+d_{Y} −d_{S}}(O_{X×Y}),

and we also consider the associated (D_{Y} , D_{X})-bimodule:

B_{S|X×Y}^{(d_{X},0)} = q_{1}^{−1}Ω_{X} ⊗_{q_{1}^{−1}O_{X}} B_{S|X×Y} .

Proposition 2.12 Assume (2.24). Then:

(i) There is a natural isomorphism of (D_{Y} , D_{X})-bimodules on S:

D_{Y ←S} ⊗_{D_{S}}^{L} D_{S→X} ⟶^{∼} B_{S|X×Y}^{(d_{X},0)} . (2.27)

In particular D_{Y ←S} ⊗_{D_{S}}^{L} D_{S→X} is concentrated in degree zero.

(ii) For M ∈ D^{b}(D_{X}), the following isomorphism holds:

\underline{Φ}_{S}M ≃ Rq_{2}_{!}(B_{S|X×Y}^{(d_{X},0)} ⊗_{q_{1}^{−1}D_{X}}^{L} q_{1}^{−1}M). (2.28)

Proof: (i) Denote by Δ_{f} the graph of f in S × X, and set Δ̃_{f} = r(Δ_{f}) ⊂ X × S. Then D_{S→X} ≃ B_{_{Δ_{f}}|X×S}^{(d_{X},0)}. Consider the diagram, where g̃ = id × g

\begin{matrix}X × S  w _{g} \\ X × Y \end{matrix} ; S^{u}.^{Δ̃_{f}  w^{∼} u^{i}} (2.29)

Then

D_{Y ←S} ⊗_{D_{S}}^{L} D_{S→X} ⊗_{O_{X}} Ω_{X}^{⊗−1} ≃ D_{X×Y ←X×S} ⊗_{D_{X×S}}^{L} B_{_{Δ_{f}}|X×S}
≃ \underline{g̃}_{∗}B_{_{Δ_{f}}|X×S}
⟶^{∼} B_{S|X×Y} ,

where the last isomorphism comes from the fact that g̃ induces an isomorphism

Δ̃_{f} ≃ S, and that B_{_{Δ_{f}}|X×S} = \underline{i}_{∗}O_{Δ_{f}}.

(ii) is an immediate consequence of (i). □

According to Proposition 2.12, a natural generalization of the transform \underline{Φ}_{S} is obtained if one replaces B_{S|X×Y}^{(d_{X},0)} by K^{(d_{X},0)} in formula (2.28), where K is a holonomic module on X × Y. Set:

K = RHom_{D_{X×Y}} (K, O_{X×Y}).

Definition 2.13 For G ∈ D^{b}(Y) and M ∈ D^{b}(D_{X}), set:

Φ_{K}G = Rq_{1}_{!}(K ⊗ q_{2}^{−1}G)[d_{Y}],
\underline{Φ}_{K}M = Rq_{2}_{!}(K^{(d_{X},0)} ⊗_{q_{1}^{−1}D_{X}}^{L} q_{1}^{−1}M).

Let Λ = char(K) be the characteristic variety of K. In order to deal with these generalized transforms, one has to replace the assumptions (2.23) and (2.24) by the following:

(2.30) the projection q_{2} : π(Λ) ⟶ Y is proper, and Λ ∩ (T^{∗}X × T_{Y}^{∗}Y) ⊂ T_{X×Y}^{∗} (X × Y).

It is possible to state our results in this more general framework, but we will not develop this approach here.

3 Vanishing theorems

3.1 E-modules

Let X be a complex manifold. We denote by π : T^{∗}X ⟶ X its cotangent bundle, by T_{X}^{∗}X the zero section of T^{∗}X, and we set:

Ṫ^{∗}X = T^{∗}X \ T_{X}^{∗}X.

If M ⊂ X is a closed submanifold, we denote by T_{M}^{∗}X its conormal bundle.

We refer to [23], [15] (see [24] for a detailed exposition) for the theory of modules over the ring E_{X} of finite order microdifferential operators on T^{∗}X.

If M is a D_{X}-module, we set:

EM = E_{X} ⊗_{π^{−1}D_{X}} π^{−1}M.

Recall that M ≃ EM|_{T_{X}^{∗}X}, and that if M is coherent, its characteristic variety, denoted char(M), is the support of EM.

Let U be a subset of T^{∗}X. We denote by Mod_{coh}(E_{X}|_{U}) the category of coherent E_{X}-modules on U, by D^{b}(E_{X}|_{U}) the full triangulated subcategory of the derived category of E_{X}|_{U}-modules whose objects have bounded cohomology, by D_{coh}^{b}(E_{X}|_{U}) the full triangulated subcategory of D^{b}(E_{X}|_{U}) whose objects have coherent cohomology.

To f : S ⟶ X one associates the maps

T^{∗}S ⟵_{^{t}f′} S ×_{X} T^{∗}X ⟶_{f_{π}} T^{∗}X.

We will denote by \underline{f}_{E}^{−1} and \underline{f}_{∗}^{E} the inverse and direct images in the sense of E-modules. Hence, for M ∈ D^{b}(E_{X}) and P ∈ D^{b}(E_{S}):

\underline{f}_{E}^{−1}M = R^{t}f′_{∗}(E_{S→X} ⊗_{f_{π}^{−1}E_{X}}^{L} f_{π}^{−1}M), \underline{f}_{∗}^{E}P = Rf_{π}_{∗}(E_{X←S} ⊗_{^{t}f^{′−1}E_{S}}^{L} ^{t}f^{′−1}P),

where E_{S→X} and E_{X←S} are the transfer bimodules.

Recall the correspondence (2.16), and assume (2.23), (2.24). Setting

Λ = T_{S}^{∗}(X × Y) ∩ (Ṫ^{∗}X × Ṫ^{∗}Y),

we consider the associated “microlocal correspondences”:

\begin{matrix}T_{S}^{∗}(X × Y)_{[ [ [ ^}_{p_{1}|_{T^{∗}_{S}(X×Y )}} _{’ ’ ’)}_{p_{2}^{a}|_{T^{∗}_{S}(X×Y )}} \\ T^{∗}X & T^{∗}Y, \end{matrix} Λ_{N N N Q}_{p_{1}|_{Λ}} _{ȷȷȷ fi}_{p_{2}^{a}|_{Λ}}

Ṫ^{∗}X  Ṫ^{∗}Y,

(3.1)

where p_{1} and p_{2} denote the projections on T^{∗}(X ×Y) ≃ T^{∗}X ×T^{∗}Y, and p_{2}^{a} = a∘p_{2}, where a denotes the antipodal map. Note that Λ = Ṫ_{S}^{∗}(X × Y) if and only if f and g are smooth.

If A is a conic subset of T^{∗}X, we set

Φ_{S}^{μ}(A) = p_{2}^{a}(T_{S}^{∗}(X × Y) ∩ p_{1}^{−1}(A)). (3.2)

We denote by C_{S|X×Y} the holonomic E_{X×Y} -module associated to S:

C_{S|X×Y} = EB_{S|X×Y} ,

and we consider the associated (E_{Y} , E_{X})-bimodule:

C_{S|X×Y}^{(d_{X},0)} = π^{−1}q_{1}^{−1}Ω_{X} ⊗_{π^{−1}q_{1}^{−1}O_{X}} C_{S|X×Y} . (3.3)

Definition 3.1 We define the functors from D^{b}(E_{X}) to D^{b}(E_{Y}):

\underline{Φ}_{S}^{μ}(M) = \underline{g}_{∗}^{E} \underline{f}_{E}^{−1}M, \underline{Ψ}_{S}^{μ}(M) = \underline{Φ}_{S}^{μ}(M)[d_{Y} − d_{X}].

Hence, we get the functors from D^{b}(E_{Y}) to D^{b}(E_{X}):

\underline{Φ}μ ; _{S}(N) = \underline{f}_{∗}^{E} \underline{g}_{E}^{−1}N , \underline{Ψ}μ ; _{S}(N) = \underline{Φ}μ ; _{S}(N)[d_{X} − d_{Y}].

We identify S ×_{X} T^{∗}X to T ∗ ; _{Δ_{f}} (X × S) and S ×_{Y} T^{∗}Y to T_{Δ_{g}}^{∗}(S × Y), and we consider the diagram

\begin{matrix}T^{∗}(X × S × Y)  w^{p_{13}} \\ T^{∗}(X × Y) \end{matrix} ; \begin{matrix}T ∗ ; _{Δ_{f}} (X × S) ×_{T^{∗}S} T_{Δ_{g}}^{∗}(S × Y)  w^{∼}^{u} \\ T_{S}^{∗}(X × Y),^{u} \end{matrix}

where p_{13} denotes as usual the natural projection defined on a product of three factors.

In the next proposition we shall write for example E_{Y ←S} instead of p_{23}^{−1}E_{Y ←S}, for short.

Proposition 3.2 Assume (2.24). Then:

(i) There is a natural isomorphism of (E_{Y} , E_{X})-bimodules on T_{S}^{∗}(X × Y):

E_{Y ←S} ⊗_{E_{S}}^{L} E_{S→X} ⟶^{∼} C_{S|X×Y}^{(d_{X},0)} .

(ii) For M ∈ D^{b}(E_{X}), the following isomorphism holds:

\underline{Φ}_{S}^{μ}(M) ≃ Rp_{2}^{a}_{∗}(C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}}^{L} p_{1}^{−1}M).

Proof: (i) We shall follow the notations of diagram (2.29). We have the chain of isomorphisms:

E_{Y ←S} ⊗_{E_{S}}^{L} E_{S→X} ⊗_{π^{−1}O_{X}} π^{−1}Ω_{X}^{⊗−1} ≃ E_{X×Y ←X×S} ⊗_{E_{X×S}}^{L} C_{_{Δ_{f}}|X×S}
= \underline{g̃}_{∗}^{E}C_{_{Δ_{f}}|X×S}
⟶^{∼} C_{S|X×Y} ,

where we have used the isomorphism E_{S→X} ≃ C_{_{Δ_{f}}|X×S}^{(d_{X},0)}, and where the last iso-

morphism follows from the fact that g̃ induces an isomorphism between Δ̃_{f} and S (see [23]).

(ii) follows from (i). □

The next result will play a crucial role in the rest of the paper.

Proposition 3.3 (i) Let f : S ⟶ X, and let M ∈ D_{coh}^{b}(D_{X}). Assume that f is non-characteristic for M. Then

E\underline{f}^{−1}M ≃ \underline{f}_{E}^{−1}EM.

(ii) Let g : S ⟶ Y, and let P ∈ D_{good}^{b}(D_{S}). Assume that g is proper on supp P.

Then

E\underline{g}_{∗}P ≃ \underline{g}_{∗}^{E}EP.

Proof: (i) is proved in [23].

(ii) was obtained in [14] in the projective case, then extended to the general case in [12], [25]. □

Corollary 3.4 Assume (2.23), (2.24), and let M ∈ D_{good}^{b}(D_{X}). Then

E(\underline{Φ}_{S}M) ≃ \underline{Φ}_{S}^{μ}(EM).

3.2 Vanishing theorems

In this section, we shall state some vanishing theorems for the cohomology of \underline{Φ}_{S}(M), M being a good D_{X}-module, making the following hypothesis:

(3.4) the map p_{2}^{a}|_{Λ} : Λ ⟶ Ṫ^{∗}Y is finite.

Note that hypotheses (2.23), (2.24), and (3.4) imply that g is open.

Proposition 3.5 Assume (2.23), (2.24), and (3.4).

(i) Let M ∈ D_{good}^{b}(D_{X}). Then char(\underline{Φ}_{S}(M)) ⊂ Φ_{S}^{μ}(char(M)).

(ii) Assume (3.4), and M ∈ Mod_{good}(D_{X}) (i.e. M is in degree 0). Then for j ≠ 0, H^{j}(\underline{Φ}_{S}M) is a holomorphic vector bundle endowed with a flat connection.

Proof: (i) is an obvious consequence of classical results on the operations on Dmodules (see [15], [24]).

(ii) It is well known that a D_{Y} -module whose characteristic variety is contained in the zero section is a locally free O_{Y} -module of finite rank endowed with a flat connection. Hence, by Corollary 3.4 it is enough to prove that:

H^{j}(\underline{Φ}_{S}^{μ}EM)|_{Ṫ^{∗}Y} = 0 for j ≠ 0.

This is clear since f being smooth \underline{f}_{E}^{−1} is exact, and p_{2}^{a}|_{Λ} being finite \underline{g}_{∗}^{E} is exact (see [23] or [24, Ch. II, Theorem 3.4.4]). □

Proposition 3.6 Assume (2.23), (2.24), (3.4), and that Y is connected. Let F be a holomorphic vector bundle on X, and recall that we set DF = D_{X} ⊗_{O_{X}} F. Then:

(i) H^{j}(\underline{Φ}_{S}DF) = 0 for j < 0,

(ii) \underline{Φ}_{S}DF is concentrated in degree zero if and only if there exists y ∈ Y such that H^{j}(ŷ; F^{∗}) = 0 for every j < d_{S/Y},

(iii) we have the isomorphism:

H^{j}(RHom_{D_{Y}} (\underline{Φ}_{S}DF, O_{Y})) ≃ Hom_{D_{Y}} (H^{−j}(\underline{Φ}_{S}DF), O_{Y}) for j ≤ 0,

(iv) if g is smooth, we have:

H^{j}(RHom_{D_{Y}} (\underline{Φ}_{S}DF, O_{Y})) = 0 for j > 0.

Proof: (i) follows from Proposition 2.12 (ii), since DF is flat over D_{X}.

(ii) Recall that Y is connected, and let y ∈ Y. Set N = \underline{Φ}_{S}DF for short, and consider the distinguished triangle

H^{0}(N) ⟶ N ⟶ τ^{>0}N ⟶_{+1},

which gives rise to the distinguished triangle

Sol(τ^{>0}N) ⟶ Sol(N) ⟶ Sol(H^{0}(N)) ⟶_{+1} (3.5)

(where, as usual, Sol(N) = RHom_{D_{Y}} (N , O_{Y})). Notice that since char(τ^{>0}N) ⊂ T_{Y}^{∗}Y, one has H^{j}Sol(τ^{>0}N) = 0 ∀j ≥ 0, (3.6)

and

τ^{>0}N = 0 ⇔ Sol(τ^{>0}N) = 0
⇔ H^{j}Sol(τ^{>0}N) = 0 ∀j < 0
⇔ H^{j}Sol(τ^{>0}N)_{y} = 0 ∀j < 0
⇔ H^{j}Sol(N)_{y} = 0 ∀j < 0,

where the last equivalence comes from the distinguished triangle (3.5), and the fact that H^{j}Sol(H^{0}(N)) = 0 for j < 0. To conclude, it remains to apply the germ formula

H^{j}Sol(N)_{y} ≃ H^{d_{S/Y} +j}(ŷ; F^{∗}).

(iii) For j < 0 we have the sequence of isomorphisms:

H^{j}Sol(N) ≃ H^{j}Sol(τ^{>0}N)
≃ Sol(H^{−j}(τ^{>0}N))
≃ Sol(H^{−j}(N)).

For j = 0 the result follows from (3.5), (3.6).

(iv) One has H^{j}(RHom_{D_{Y}} (N , O_{Y}))_{y} ≃ H^{d_{S/Y} +j}(ŷ; F^{∗}) and this group is zero since ŷ ≃ g^{−1}(y) is a compact smooth submanifold of X of dimension d_{S/Y}. In fact, if Z is a compact smooth submanifold of dimension d of the complex manifold X, and F is a locally free O_{X}-module of finite rank on X, the vanishing of H^{d+j}(Z; F) for j > 0 follows by duality from the fact that H_{Z}^{k}(X; F^{∗} ⊗_{O_{Y}} Ω_{Y}) is zero for k < d_{X} −d, and for k = d_{X} − d this space is isomorphic to Γ(X; H_{Z}^{d_{X}−d}(F^{∗} ⊗_{O_{Y}} Ω_{Y})), hence has a natural topology of a (separated) Fréchet space. □

4 A regularity theorem and an equivalence of categories

4.1 Modules with regular singularities

We review some notions and results from Kashiwara and Oshima’s work [16].

The ring E_{X} is naturally endowed with a ℤ-filtration by the degree, and we denote by E_{X}(k) the sheaf of operators of degree at most k. Denote by O_{T^{∗}X}(k) the sheaf of holomorphic functions on T^{∗}X, homogeneous of degree k.

Let V ⊂ Ṫ^{∗}X be a conic regular involutive submanifold of codimension c_{V}. Recall that one says that a smooth conic involutive manifold V is regular if the canonical one-form on T^{∗}X does not vanish on V. Denote by I_{V} (k) the sheaf ideal of sections of O_{T^{∗}X}(k) vanishing on V. Let E_{V} be the subalgebra of E_{X} generated over E_{X}(0) by the sections P of E_{X}(1) such that σ_{1}(P) belongs to I_{V} (1) (here σ_{1}(·) denotes the symbol of order 1).

Example 4.1 Let X = W × Z, V = U_{W} × T_{Z}^{∗}Z for an open subset U_{W} ⊂ Ṫ^{∗}W. Then E_{V} is the subalgebra D_{Z}E_{X}(0) of E_{X} generated over E_{X}(0) by the differential operators of Z.

Definition 4.2 (cf [16]). Let M be a coherent E_{X}-module. One says that M has regular singularities along V if locally there exists a coherent sub-E_{X}(0)-module M_{0} of M which generates it over E_{X}, and such that E_{V} M_{0} ⊂ M_{0}. One says that M is simple along V if locally there exists an E_{X}(0)-module M_{0} as above such that M_{0}/E_{X}(−1)M_{0} is a locally free O_{V} (0)-module of rank one.

Notice that the above definitions are invariant by quantized contact transformations, and that a system with regular singularities along V is supported by V (cf. [16, Lemma 1.13]).

We will denote by Mod_{RS(V)}(E_{X}) the thick abelian subcategory of Mod_{coh}(E_{X}) whose objects have regular singularities along V. We denote by D_{RS(V)}^{b}(E_{X}) the full triangulated subcategory of D_{coh}^{b}(E_{X}) whose objects M have cohomology groups with regular singularities along V. This category is invariant by quantized contact transformations.

Example 4.3 Let V ⊂ Ṫ^{∗}X be a regular involutive submanifold, and let S_{V} be simple along V. We may locally assume, after a quantized contact transformation, that X = W × Z, V = U_{W} × T_{Z}^{∗}Z. In this case S_{V} is isomorphic to the partial de Rham system E_{W} \underline{×} O_{Z} (cf. [16, Theorem 1.9]). Denoting by ρ : V ⟶ U_{W} the natural projection, we notice that End_{E_{X}}(S_{V}) ≃ ρ^{−1}E_{W}, which shows in particular that the ring End_{E_{X}}(S_{V}) of E_{X}-linear endomorphisms of S_{V} is coherent.

There are useful criterion to ensure that an E_{X}-module M has regular singularities on V.

Proposition 4.4 Let V ⊂ Ṫ^{∗}X be a regular involutive submanifold, let M be a coherent E_{X}-module, and let S_{V} be simple along V. Then M has regular singularities along V if and only if for any d > 0 there locally exists an exact sequence of E_{X}- modules: S_{V}^{N_{d}} ⟶ ⋯ ⟶ S_{V}^{N_{0}} ⟶ M ⟶ 0. (4.1)

Proof: We may assume that X = W × Z, with dim Z = d_{Z} = c_{V}, and V = U_{W} × T_{Z}^{∗}Z, U_{W} being open in Ṫ^{∗}W.

Assume that M has regular singularities. Let M_{0} be a coherent E_{X}(0)-module which generates M, such that E_{V} M_{0} ⊂ M_{0}, and let u_{1}, …, u_{r} be a system of generators of M_{0}. Let (y_{1}, …, y_{d_{Z}}) be a local coordinate system on Z. Then there exist r × r matrices A_{j}, j = 1, …, c_{V}, with entries in E_{X}(0), such that:

D_{y_{j}} ( u_{1} ; ⋮ ; u_{r} ) = A_{j} ( u_{1} ; ⋮ ; u_{r} ) (4.2)

Denote by M′ the E_{X}-module with generators v_{1}, …, v_{r}, and relations (4.2). Then the map v_{j} ↦ u_{j} defines the E_{X}-linear exact sequence:

M′ ⟶_{ψ} M ⟶ 0.

Set M′_{0} = E_{X}(0)v_{1} + ⋯ + E_{X}(0)v_{r}. By the above relations, M′_{0}/E_{X}(−1)M′_{0} is locally free over O_{V} (0). Hence M′ is locally isomorphic to S_{V}^{N_{0}} for some N_{0} (see [16, Theorem 1.9]). Let M_{1} be the kernel of ψ. Then M_{1} has regular singularities on V, and the induction proceeds.

Conversely, assume (4.1). Then the fact that M has regular singularities is a consequence of the fact that Mod_{RS(V)}(E_{X}) is a thick subcategory of Mod_{coh}(E_{X}). □

Remark 4.5 Let us denote by Car_{V}^{1} (M) the 1-micro-characteristic variety of a coherent E_{X}-module, introduced by Y. Laurent [18] and T. Monteiro-Fernandes [20] (see also [24, p. 123]). Then one can show that M has regular singularities along V if and only if Car_{V}^{1} (M) ⊂ V, the zero section of T_{V} T^{∗}X.

Proposition 4.6 Let V ⊂ Ṫ^{∗}X be a regular involutive submanifold, and let S_{V} be simple along V. Set A_{V} = End_{E_{X}}(S_{V}). Then the two functors:

Mod_{RS(V)}(E_{X})  _{w}_{α} Mod_{coh}(A_{V})^{u} ^{β}

given by:

α(M) = Hom_{E_{X}}(S_{V} , M), β(R) = S_{V} ⊗_{A_{V}} R,

are well-defined and quasi-inverse to each other. Moreover, M ∈ Mod_{RS(V)}(E_{X}) is simple along V if and only if α(M) is a locally free A_{V} -module of rank one.

Proof: (i) Let us show that α is well-defined. First, notice that for any coherent E_{X}-module M:

Ext_{E_{X}}^{j}(S_{V} , M) = 0, for j > c_{V} .

For M in Mod_{RS(V)}(E_{X}), consider an E_{X}-linear exact sequence:

0 ⟶ Z^{d} ⟶ S_{V}^{N_{d}} ⟶ ⋯ ⟶ S_{V}^{N_{0}} ⟶ M ⟶ 0. (4.3)

Arguing by induction, we find that Z^{d} has regular singularities on V. Moreover, by standard arguments, we get that

Ext_{E_{X}}^{j}(S_{V} , M) = 0, for j > 0.

Hence Hom_{E_{X}}(S_{V} , ·) is exact on Mod_{RS(V)}(E_{X}). Applying Hom_{E_{X}}(S_{V} , ·) to (4.3), we thus get an A_{V} -linear resolution:

A_{V}^{N_{1}} ⟶ A_{V}^{N_{0}} ⟶ α(M) ⟶ 0.

(ii) β is well defined. In fact, it is enough to check it for R = A_{V}, that is, to check that S_{V} has regular singularities along V, which is clear.

(iii) id ⟶^{∼} α ∘ β. In fact, let R ∈ Mod_{coh}(A_{V}). Then:

R ⟶^{∼} Hom_{E_{X}}(S_{V} , S_{V}) ⊗_{A_{V}} R
⟶^{∼} Hom_{E_{X}}(S_{V} , S_{V} ⊗_{A_{V}} R)
= α ∘ β(R).

(iv) β ∘ α ⟶^{∼} id. In fact, let M ∈ Mod_{RS(V)}(E_{X}). To check that the natural morphism: S_{V} ⊗_{A_{V}} Hom_{E_{X}}(S_{V} , M) ⟶ M (4.4)

is an isomorphism, we may proceed locally and use Proposition 4.4. Consider a resolution: S_{V}^{N_{1}} ⟶ S_{V}^{N_{0}} ⟶ M ⟶ 0. (4.5)

We have already noticed that the sequence:

α(S_{V}^{N_{1}}) ⟶ α(S_{V}^{N_{0}}) ⟶ α(M) ⟶ 0

remains exact. The functor β being right exact, the sequence:

β(α(S_{V}^{N_{1}})) ⟶ β(α(S_{V}^{N_{0}})) ⟶ β(α(M)) ⟶ 0

is exact. Since β(α(S_{V})) ⟶^{∼} S_{V} this last sequence is but the exact sequence of E_{X}-modules: S_{V}^{N_{1}} ⟶ S_{V}^{N_{0}} ⟶ β(α(M)) ⟶ 0. (4.6)

Comparing (4.5) and (4.6) we get (4.4) by the five lemma. □

4.2 A regularity theorem

Let V ⊂ Ṫ^{∗}X be a conic regular involutive submanifold. We say that a coherent D_{X}-module M has regular singularities on V, if so has EM. We denote by Mod_{RS(V)}(D_{X}) the thick subcategory of Mod_{good}(D_{X}) whose objects have regular singularities on V, and by D_{RS(V)}^{b}(D_{X}) the full triangulated subcategory of D_{good}^{b}(D_{X}) whose objects have cohomology groups belonging to Mod_{RS(V)}(D_{X}).

Recall the correspondence (2.16), and the associated microlocal correspondence (3.1):

Λ_{N N N Q}_{p_{1}|_{Λ}} _{ȷȷȷ fi}_{p_{2}^{a}|_{Λ}}

Ṫ^{∗}X  Ṫ^{∗}Y.

The manifold Λ being Lagrangian, it is well known that p_{1}|_{Λ} is smooth if and only if p_{2}^{a}|_{Λ} is an immersion. We will assume:

p_{2}^{a}|_{Λ} is a closed embedding identifying Λ to a closed regular involutive submanifold V ⊂ Ṫ^{∗}Y, and p_{1}|_{Λ} is smooth and surjective on Ṫ^{∗}X.

(4.7)

Let us denote by c_{V} the complex codimension of V in T^{∗}Y. We have the following local model for the correspondence (3.1).

Lemma 4.7 Assume (2.24), (4.7). Then, for every (p, q^{a}) ∈ Λ there exist open subsets U_{X}, U′_{X} ⊂ Ṫ^{∗}X, U_{Y} ⊂ Ṫ^{∗}Y, with p ∈ U_{X} and q ∈ U_{Y}, a complex manifold Z of dimension c_{V}, and a contact transformation ψ : U_{Y} ⟶^{∼} U′_{X} × T^{∗}Z, such that id_{U_{X}} × ψ induces an isomorphism of correspondences:

\begin{aligned}& Λ ∩ (U_{X} × U_{Y}^{a})_{[ [ [ ^}_{p_{1}}  _{’ ’ ’)}_{p_{2}^{a}} \\ U_{X} & & V ∩ U_{Y} \end{aligned} ⟶^{∼} \begin{aligned}& Λ_{χ} × T_{Z}^{∗}Z_{[ [ [ ^}_{p_{1}}  _{’ ’ ’)}_{p_{23}^{a}} \\ U_{X} & U′_{X} × T_{Z}^{∗}Z, \end{aligned}

where Λ_{χ} ⊂ U_{X} × U′_{X}^{a} is the graph of a contact transformation χ : U_{X} ⟶^{∼} U′_{X}, and p_{23}^{a} denotes the projection U_{X} × U′_{X}^{a} × T^{∗}Z ⟶ U′_{X} × T^{∗}Z.

Proof: Since V is regular involutive, there exist complex manifolds X′ and Z of dimension d_{Y} − c_{V} and c_{V} respectively, open subsets U′_{X} ⊂ Ṫ^{∗}X′, U_{Y} ⊂ Ṫ^{∗}Y with p ∈ U_{Y}, and a contact transformation ψ : U_{Y} ⟶^{∼} U′_{X} × T^{∗}Z such that

ψ(V ∩ U_{Y}) = U′_{X} × T_{Z}^{∗}Z.

By hypothesis (4.7), one has Λ ⊂ Ṫ^{∗}X × V. In particular, Λ is invariant by the bicharacteristic flow of Ṫ^{∗}X × V, and hence id_{Ṫ^{∗}X} × ψ interchanges Λ ∩ (Ṫ^{∗}X × U_{Y}) with Λ′ × T_{Z}^{∗}Z, for a Lagrangian manifold Λ′ of Ṫ^{∗}X × U′_{X}. Since p_{2}^{a}|_{Λ} gives an isomorphism Λ ⟶^{∼} V, we also have Λ′ ⟶^{∼} U′_{X}. The manifold Λ′ being Lagrangian, there exists an open subset U_{X} ⊂ Ṫ^{∗}X such that Λ′ = Λ_{χ} is the graph of a contact transformation χ : U_{X} ⟶^{∼} U′_{X}. In particular, d_{Y} − c_{V} = d_{X}, and it is not restrictive to assume that X′ = X. □

Theorem 4.8 Assume (2.23), (2.24) and (4.7).

(i) If M ∈ D_{good}^{b}(D_{X}), then \underline{Φ}_{S}(M) belongs to D_{RS(V)}^{b}(D_{Y}).

(ii) If M ∈ Mod_{good}(D_{X}), and M is locally free of rank one, then H^{0}(\underline{Φ}_{S}(M)) is simple along V on Ṫ^{∗}Y.

Proof: By (4.7), the problem is local on Λ. Hence by “dévissage”, we may assume M = D_{X}. Since the transform \underline{Φ}_{S} “commutes to microlocalization” (Corollary 3.4), it is enough to show that p_{2}^{a}_{∗}C_{S|X×Y}^{(d_{X},0)} has regular singularities on V. This statement is invariant by quantized contact transformations on Y, and we may apply Lemma 4.7. Hence, it is enough to show that if S is a simple holonomic E_{X×Y} -module on Λ = Λ_{χ} × T_{Z}^{∗}Z, where Λ_{χ} ⊂ U_{X} × U′_{X}^{a} is the graph of a contact transformation, then p_{23}^{a}_{∗}S^{(d_{X},0)} is a simple system on U′_{X} × T_{Z}^{∗}Z. This is obvious, since locally S^{(d_{X},0)} ≃ S_{χ}^{(d_{X},0)} \underline{×} O_{Z}, S_{χ} being a simple holonomic system on Λ_{χ}, and S_{χ}^{(d_{X},0)} is isomorphic (as an E_{X}-module) to E_{X}. □

4.3 An equivalence of categories

In this paragraph we shall prove that the transform \underline{Φ}_{S} “almost” induces an equivalence between the category of coherent D-modules on X, and the category of coherent D-modules on Y with regular singularities on V.

Here, we will make the assumptions below:

(4.8) f and g are smooth and proper, (4.9) (f, g) : S ↪ X × Y is a closed embedding, (4.10) f has connected and simply connected fibers,

p_{2}^{a}|_{Λ} is a closed embedding identifying Λ to a closed regular involutive submanifold V ⊂ Ṫ^{∗}Y, and p_{1}|_{Λ} is smooth and surjective on Ṫ^{∗}X.

(4.11)

We define the category Mod_{good}(D_{X}; Ṫ^{∗}X) as the localization of Mod_{good}(D_{X}) by the thick subcategory of holomorphic vector bundles endowed with a flat connection:

N_{X} = {M ∈ Mod_{good}(D_{X}); char(M) ⊂ T_{X}^{∗}X}.

In particular, the objects of Mod_{good}(D_{X}; Ṫ^{∗}X) are the same as the objects of Mod_{good}(D_{X}), and a morphism u : M ⟶ M′ in Mod_{good}(D_{X}) becomes an isomorphism in Mod_{good}(D_{X}; Ṫ^{∗}X) if ker u and coker u belong to N_{X}. This is equivalent to say that Eu : EM ⟶ EM′ is an isomorphism on Ṫ^{∗}X. We similarly define N_{Y}, and the category Mod_{RS(V)}(D_{Y}; Ṫ^{∗}Y) obtained by localizing Mod_{RS(V)}(D_{Y}) with respect to N_{Y}.

Taking the zero-th cohomology groups of the functors \underline{Φ}_{S} and \underline{Ψ}_{S}, we get functors that we will denote by \underline{Φ}_{S}^{0} and \underline{Ψ}0 ; _{S}. In other words, we set:

\underline{Φ}_{S}^{0} = H^{0} ∘ \underline{Φ}_{S}, \underline{Ψ}0 ; _{S} = H^{0} ∘ \underline{Ψ}_{S}.

Hence (using Theorem 4.8) we get the functors:

Mod_{good}(D_{X})  _{w}_{\underline{Φ}_{0}_{S}} Mod_{RS(V)}(D_{Y}).^{u} ^{\underline{Ψ}0 ; _{S}}

These functors interchange N_{X} and N_{Y}, and hence induce the functors, that we denote by \underline{Φ}_{S}^{0·} and \underline{Ψ}0· ; _{S}:

Mod_{good}(D_{X}; Ṫ^{∗}X)  _{w}_{\underline{Φ}{0· \atop S}} Mod_{RS(V)}(D_{Y}; Ṫ^{∗}Y).^{u} ^{\underline{Ψ}0· ; _{S}}

(4.12)

Theorem 4.9 Assume (4.8)–(4.11). Then the functors \underline{Φ}_{S}^{0·} and \underline{Ψ}0· ; _{S} are quasi-inverse to each other, hence define an equivalence of categories.

Recall that by Proposition 2.6 the functors \underline{Φ}_{S} and \underline{Ψ}_{S}:

D_{good}^{b}(D_{X})  _{w}_{\underline{Φ}_{S}} D_{RS(V)}^{b}(D_{Y}),^{u} ^{\underline{Ψ}_{S}}

are adjoint to each other. Hence we have natural morphisms:

(4.13) id ⟶ \underline{Ψ}_{S} ∘ \underline{Φ}_{S} in D_{good}^{b}(D_{X}),

and:

\underline{Φ}_{S} ∘ \underline{Ψ}_{S} ⟶ id in D_{RS(V)}^{b}(D_{Y}). (4.14)

Let M ∈ D_{good}^{b}(D_{X}), N ∈ D_{RS(V)}^{b}(D_{Y}). By Corollary 3.4, we get morphisms:

(4.15)  EM ⟶ \underline{Ψ}μ ; _{S}(\underline{Φ}_{S}^{μ}(EM)) in D^{b}(E_{X}|_{Ṫ^{∗}X}),
(4.16)  \underline{Φ}_{S}^{μ}(\underline{Ψ}μ ; _{S}(EN)) ⟶ EN  in D^{b}(E_{Y} |_{Ṫ^{∗}Y}).

Recall that

\underline{Φ}_{S}^{μ}(M) ≃ Rp_{2}^{a}_{∗}(C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}}^{L} p_{1}^{−1}M) for M ∈ D^{b}(E_{X}|_{Ṫ^{∗}X}),
\underline{Ψ}μ ; _{S}(N) ≃ Rp_{1}_{∗}(C_{_{S}|Y ×X}^{(d_{Y} ,0)} ⊗_{p_{2}^{a}^{−1}E_{Y}}^{L} p_{2}^{a}^{−1}N)[d_{X} − d_{Y}] for N ∈ D^{b}(E_{Y} |_{Ṫ^{∗}Y}),

where C_{S|X×Y}^{(d_{X},0)} is the module defined in (3.3).

In order to prove Theorem 4.9, we will need a few lemmas.

Lemma 4.10 On Λ = Ṫ_{S}^{∗}(X × Y), the module C_{S|X×Y}^{(d_{X},0)} is flat over p_{1}^{−1}E_{X}, and has T or-dimension d_{Y} − d_{X} over p_{2}^{a}^{−1}E_{Y}. In particular, the functor \underline{Ψ}μ ; _{S} is left exact for the natural t-structures of D^{b}(E_{Y} |_{Ṫ^{∗}Y}) and D^{b}(E_{X}|_{Ṫ^{∗}X}).

Proof: The problem is local on Λ. Applying Lemma 4.7, we may assume Y = X×Z, Λ = Λ_{χ} × T_{Z}^{∗}Z, C_{S|X×Y} ≃ C_{Λ_{χ}} × O_{Z}, where C_{Λ_{χ}} is a simple E_{X×X}-module on Λ_{χ}. The result is then clear, since d_{Z} = d_{Y} − d_{X}. □

Lemma 4.11 The functor:

\underline{Φ}_{S}^{μ} : Mod_{coh}(E_{X}|_{Ṫ^{∗}X}) ⟶ Mod_{coh}(E_{Y} |_{Ṫ^{∗}Y}),

is well defined and exact.

Proof: We have:

\underline{Φ}_{S}^{μ}(·) = Rp_{2}^{a}_{∗}(C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}}^{L} p_{1}^{−1}(·)).

Since the map p_{2}^{a} is finite on Ṫ_{S}^{∗}(X ×Y), and C_{S|X×Y}^{(d_{X},0)} is flat over p_{1}^{−1}E_{X}, the statement follows, using Proposition 3.2. □

We set:

\underline{Φ}_{S}^{μ0} = H^{0} ∘ \underline{Φ}_{S}^{μ}, \underline{Ψ}μ0 ; _{S} = H^{0} ∘ \underline{Ψ}μ ; _{S}.

Let M ∈ Mod_{good}(D_{X}) and N ∈ Mod_{RS(V)}(D_{Y}). Applying the functor H^{0}(·) to the morphisms (4.15) and (4.16), we get the following morphisms, in view of Lemmas 4.10 and 4.11:

(4.17)  EM ⟶ \underline{Ψ}μ0 ; _{S}(\underline{Φ}_{S}^{μ0}(EM)) in Mod(E_{X}|_{Ṫ^{∗}X}),
(4.18)  \underline{Φ}_{S}^{μ0}(\underline{Ψ}μ0 ; _{S}(EN)) ⟶ EN  in Mod(E_{Y} |_{Ṫ^{∗}Y}).

Lemma 4.12 The morphisms (4.17) and (4.18) are isomorphisms on Ṫ^{∗}X and Ṫ^{∗}Y respectively.

Proof: For M ∈ Mod_{coh}(E_{X}|_{Ṫ^{∗}X}) and N ∈ Mod_{RS(V)}(E_{Y} |_{Ṫ^{∗}Y}), we have:

\underline{Φ}_{S}^{μ0}(M) = p_{2}^{a}_{∗}(C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}} p_{1}^{−1}M),
\underline{Ψ}μ0 ; _{S}(EN) = p_{1}_{∗}T or_{d_{Y} −d_{X}}^{p_{2}^{a}^{−1}E_{Y}}(C_{_{S}|Y ×X}^{(d_{Y} ,0)}, p_{2}^{a}^{−1}N).

From now on, we shall not write p_{2}^{a} (which is an isomorphism) for short. (i) Let us prove that (4.17) is an isomorphism on Ṫ^{∗}X. Since p_{1} has connected and simply connected fibers by (4.10), M is isomorphic

to p_{1}_{∗}p_{1}^{−1}M, and hence it is enough to prove the isomorphism:

p_{1}^{−1}M ≃ T or_{d_{Y} −d_{X}}^{E_{Y}}(C_{_{S}|Y ×X}^{(d_{Y} ,0)}, C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}} p_{1}^{−1}M).

This is a local problem on Λ. Applying Lemma 4.7, the correspondence:

Λ_{N N N Q}_{p_{1}} _{ȷȷȷ fi}_{p_{2}^{a}}

Ṫ^{∗}X  V,

is locally isomorphic to the correspondence:

\begin{aligned}& Λ_{χ} × T_{Z}^{∗}Z_{[ [ [ ^}_{p_{1}} _{’ ’ ’)}_{p_{23}^{a}} \\ U_{X} & U′_{X} × T_{Z}^{∗}Z. \end{aligned}

Let C_{Λ_{χ}} be a simple holonomic E_{X×X}-module on Λ_{χ}. Then, locally on Λ one has an isomorphism:

C_{_{S}|Y ×X} ≃ C_{Λ_{χ}} \underline{×} O_{Z}

as an E_{Y ×X}-module, and:

C_{_{S}|Y ×X}^{(d_{Y} ,0)} ⊗_{E_{Y}}^{L} (C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}} p_{1}^{−1}M)[d_{X} − d_{Y}]

≃ (C_{Λ_{χ}}^{(d_{X},0)} \underline{×} Ω_{Z}) ⊗_{E_{X×Z}}^{L} (C(d_{X},0) ; _{Λ_{χ}} \underline{×} O_{Z}) ⊗_{p_{1}^{−1}E_{X}} p_{1}^{−1}M[d_{X} − d_{Y}]
≃ (C_{Λ_{χ}}^{(d_{X},0)} ⊗_{E_{X}}^{L} C(d_{X},0) ; _{Λ_{χ}}) ⊗_{p_{1}^{−1}E_{X}} p_{1}^{−1}M
≃ p_{1}^{−1}M,

where we set C_{Λ_{χ}}^{(d_{X},0)} = C_{Λ_{χ}} ⊗_{q_{2}^{−1}O_{X}} q_{2}^{−1}Ω_{X}, and we similarly define C(d_{X},0) ; _{Λ_{χ}} for Λ̃_{χ} = r(Λ_{χ}) (here q_{2} is the second projection X × X ⟶ X). To prove the above isomorphisms we have used the fact that:

Ω_{Z} ⊗_{D_{Z}}^{L} O_{Z}[d_{X} − d_{Y}] ≃ ℂ_{Z}

(recall that d_{Y} − d_{X} = d_{Z}), and:

C_{Λ_{χ}}^{(d_{X},0)} ⊗_{E_{X}}^{L} C(d_{X},0) ; _{Λ_{χ}} ≃ E_{X},

which holds since Λ_{χ} is the graph of a contact transformation on Ṫ^{∗}X. (ii) Let us prove that (4.18) is an isomorphism. One has:

\underline{Φ}_{S}^{μ0}(\underline{Ψ}μ0 ; _{S}(EN)) ≃ C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}} p_{1}^{−1}p_{1}_{∗}T or_{d_{Y} −d_{X}}^{p_{2}^{a}^{−1}E_{Y}}(C_{_{S}|Y ×X}^{(d_{Y} ,0)}, N).

Set:

Q = T or_{d_{Y} −d_{X}}^{p_{2}^{a}^{−1}E_{Y}}(C_{_{S}|Y ×X}^{(d_{Y} ,0)}, N).

Let us show that Q is locally constant along the fibers of p_{1}. With the same notations

as in (i), one may assume N ≃ N′ \underline{×} O_{Z}, C_{_{S}|Y ×X}^{(d_{Y} ,0)} ≃ C_{Λ_{χ}}^{(d_{X},0)} \underline{×} O_{Z}. Hence

Q ≃ H^{0}((C_{Λ_{χ}}^{(d_{X},0)} \underline{×} Ω_{Z}) ⊗_{E_{X×Z}}^{L} (N′ \underline{×} O_{Z})[d_{X} − d_{Y}])
≃ H^{0}(C_{Λ_{χ}}^{(d_{X},0)} ⊗_{E_{X}}^{L} N′) × ℂ_{Z}
≃ (C_{Λ_{χ}}^{(d_{X},0)} ⊗_{E_{X}} N′) × ℂ_{Z}

(note that C_{Λ_{χ}}^{(d_{X},0)} is flat over E_{X}).

Since the fibers of p_{1} are connected and simply connected, and Q is locally constant along these fibers, one has p_{1}^{−1}p_{1}_{∗}Q ⟶^{∼} Q. Whence:

\underline{Φ}_{S}^{μ0}(\underline{Ψ}μ0 ; _{S}(EN)) ≃ C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}} Q,

and to check that it is isomorphic to N is now a local problem on V, i.e. on Λ. Then

the proof goes as in (i): we may assume C_{_{S}|Y ×X}^{(d_{Y} ,0)} ≃ C_{Λ_{χ}}^{(d_{Y} ,0)} \underline{×} O_{Z}, Q ≃ (C_{Λ_{χ}}^{(d_{X},0)} ⊗_{E_{X}}^{L}

N′) × ℂ_{Z}, N ≃ N′ \underline{×} O_{Z}, and we get:

C_{S|X×Y}^{(d_{X},0)} ⊗_{p_{1}^{−1}E_{X}} Q ≃ N′ \underline{×} O_{Z}
≃ N .
□

Proof of of Theorem 4.9: Denote by D_{good}^{b}(D_{X}; Ṫ^{∗}X) the localization of D_{good}^{b}(D_{X}) by the null system:

{M ∈ D_{good}^{b}(D_{X}); char(M) ⊂ T_{X}^{∗}X}.

This is a triangulated category which inherits of the natural t-structure of D_{good}^{b}(D_{X}) and its heart for this t-structure is the category Mod_{good}(D_{X}; Ṫ^{∗}X). One has a similar construction for D_{RS(V)}^{b}(D_{X}; Ṫ^{∗}X).

Denote by \underline{Φ}_{S}^{·} and \underline{Ψ}· ; _{S} the image of \underline{Φ}_{S} and \underline{Ψ}_{S} in these localized categories. The morphisms (4.13) and (4.14) define the morphisms:

id ⟶ \underline{Ψ}· ; _{S} ∘ \underline{Φ}_{S}^{·},
\underline{Φ}_{S}^{·} ∘ \underline{Ψ}· ; _{S} ⟶ id.

Moreover, \underline{Φ}_{S}^{·} is t-exact in view of Proposition 3.5 (ii). Hence we obtain the morphisms:

(4.19) id ⟶ \underline{Ψ}0· ; _{S} ∘ \underline{Φ}_{S}^{0·} in Mod_{good}(D_{X}; Ṫ^{∗}X),

\underline{Φ}_{S}^{0·} ∘ \underline{Ψ}0· ; _{S} ⟶ id in Mod_{RS(V)}(D_{Y}; Ṫ^{∗}Y), (4.20)

and we have to show that these are isomorphisms.

Let M ∈ Mod_{good}(D_{X}; Ṫ^{∗}X). Since EM ≃ 0 implies M ≃ 0, (4.19) and (4.20) are isomorphisms in view of Lemma 4.12. □

Remark 4.13 A related result to Theorem 4.9 is obtained by Brylinski [2] in the framework of perverse sheaves.

An equivalence of categories for modules with regular singularities along the manifold X ×_{Y} T^{∗}Y associated to a smooth map Y ⟶ X is obtained by Honda [11].

5 Applications

In this section we will apply our general results to the classical case of the twistor correspondence considered in the introduction. In particular, we discuss to what extent we can recover, without any explicit calculation, the results of [6], [29].

5.1 The twistor correspondence (holomorphic solutions)

Let us consider the twistor correspondence

ℙ  𝕄,^{𝔽 N N N Q_{f}  _{ȷȷȷ fi}_{g}}

(5.1)

where 𝔽 = 𝔽_{1,2}(𝕋) is the flag manifold of type (1, 2) of a four-dimensional complex vector space 𝕋 (called twistor space), ℙ = 𝔽_{1}(𝕋) is a projective three-space, and 𝕄 = 𝔽_{2}(𝕋) is identified with the four-dimensional compactified complexified Minkowski space. The projections here are given by f(L_{1}, L_{2}) = L_{1}, g(L_{1}, L_{2}) = L_{2}, where L_{1} ⊂ L_{2} ⊂ 𝕋 are complex subspaces of dimension one and two respectively, defining an element (L_{1}, L_{2}) of 𝔽.

As we said in the introduction, we denote by _{h} the massless field equation of helicity h, and, for k ∈ ℤ, we denote by O_{ℙ}(k) the −k-th tensor power of the tautological bundle on ℙ.

In order to apply to this situation our previous results, let us begin by verifying that hypotheses (4.8)–(4.11) are satisfied.

Clearly f and g are smooth and proper.

Choose local coordinates (x_{1}, x_{2}, x_{3}), (y_{1}, y_{2}, y_{3}, y_{4}) on affine charts of ℙ and 𝕄 respectively and denote by (x; ξ), (y; η) the associated coordinates on T^{∗}ℙ and T^{∗}𝕄 respectively. Here (x_{1}, x_{2}, x_{3}) corresponds to the line generated by (1, x_{1}, x_{2}, x_{3}) ∈ 𝕋 and (y_{1}, y_{2}, y_{3}, y_{4}) corresponds to the two-plane of 𝕋 generated by the vectors (y_{2}, y_{4}, 0, 1) and (y_{1}, y_{3}, 1, 0). The submanifold 𝔽 of ℙ × 𝕄 is given by the system of equations

( \begin{matrix}1 \\ x_{1} \\ x_{2} \\ x_{3} \end{matrix} ) ∧ ( \begin{matrix}y_{1} \\ y_{3} \\ 1 \\ 0 \end{matrix} ) ∧ ( \begin{matrix}y_{2} \\ y_{4} \\ 0 \\ 1 \end{matrix} ) = 0.

On the open set y_{1} ≠ 0 we find the independent equations

\{ \begin{matrix}x_{1} − x_{2}y_{3} − x_{3}y_{4} = 0, \\ 1 − x_{2}y_{1} − x_{3}y_{2} = 0. \end{matrix}

The fiber of Λ = T ∗_{𝔽}(ℙ × 𝕄) ∩ (Ṫ^{∗}ℙ × Ṫ^{∗}𝕄) at (x, y) is given by:

λdx_{1} + (−λy_{3} − μy_{1})dx_{2} + (−λy_{4} − μy_{2})dx_{3}
−μx_{2}dy_{1} − μx_{3}dy_{2} − λx_{2}dy_{3} − λx_{3}dy_{4}

for λ, μ ∈ ℂ^{×}. Then one checks that p_{2}^{a}|_{Λ} is an embedding. Set:

V = p_{2}^{a}(Λ).

Then V ∩ π−1_{𝕄}({y; y_{1} ≠ 0}) is given by the equation η_{1}η_{4} = η_{2}η_{3}, i.e. it is the characteristic variety of the wave equation (up to a ℂ-linear change of coordinates). It is then easy to check that (4.9), (4.10) and (4.11) are verified.

Lemma 5.1 For y ∈ 𝕄 one has

(i) H^{0}(ŷ; O_{ℙ}(k)) = \{ \begin{matrix}0 & for k < 0, \\ ≠ 0 & and finite dimensional for k ≥ 0, \end{matrix}

(ii) H^{1}(ŷ; O_{ℙ}(k)) is infinite dimensional for any k,

(iii) H^{j}(ŷ; O_{ℙ}(k)) = 0 for j ≠ 0, 1 and for any k.

Since this result is well known from the specialists, we do not write down the proof, which is an explicit calculation using Čech cohomology. (A detailed proof appears in [3, Correspondences for D-modules and Penrose transform].)

Hence, as a particular case of the preceding results, we obtain:

Proposition 5.2 (i) Let M be a good D_{ℙ}-module. Then H^{0}(\underline{Φ}_{𝔽}(M)) is a good D_{𝕄}-module with regular singularities on V, and H^{j}(\underline{Φ}_{𝔽}(M)) is a flat holomorphic connection for j ≠ 0. Moreover, any good D_{𝕄}-module with regular singularities on V is, up to a flat connection, the unique image by \underline{Φ}_{𝔽} of a good D_{ℙ}-module (see Theorem 4.9 for a precise statement).

(ii) Let k ∈ ℤ, and set D_{ℙ}(−k) = D_{ℙ} ⊗_{O_{ℙ}} O_{ℙ}(−k). Then H^{j}(\underline{Φ}_{𝔽}(D_{ℙ}(−k))) = 0 for j < 0, for j = 0 this module is simple along V, and it is zero for j > 0 if and only if k < 0.

(iii) For y ∈ Y one has:

RΓ(ŷ; O_{ℙ}(k)) ≃ RHom_{D_{Y}} (\underline{Φ}_{𝔽}D_{ℙ}(−k), O_{Y})_{y}[−1].

(iv) Let U ⊂ 𝕄 be a 𝔽̃-trivial open subset. Then for k < 0:

H^{j+1}(Û; O_{ℙ}(k)) ≃ Ext_{D_{𝕄}}^{j}(U; \underline{Φ}_{𝔽}D_{ℙ}(−k), O_{𝕄}).

Remark 5.3 (i) Theorem 1.1 may now be stated as follows: for k < 0, the D-module transform of D_{ℙ}(−k) is the D_{𝕄}-module associated to the equation h(k), where h(k) = −(1 + k/2). We refer to [6] for the calculations implicitly leading to this conclusion.

(ii) Eastwood et al. [6] deals with complexes of locally free O_{𝕄}-modules and D_{𝕄}- linear morphisms. This category is equivalent to that of filtered D_{𝕄}-modules as proved by Saito [22]. This shows that these authors indeed use D-module theory.

(iii) In order to get the isomorphism of Proposition 5.2 (iv), we assumed 𝔽̃-triviality of U. Since in [6] the authors were not interested in computing all cohomology groups, they could slightly weaken the topological hypothesis and only assume that U ∩ x̂ is connected and simply connected for every x ∈ Û.

5.2 The twistor correspondence (hyperfunction solutions)

Now we shall apply Proposition 2.6 to the study of hyperfunction solutions, and show how to easily recover the results of Wells [29].

Let ϕ be a Hermitian form on 𝕋 of signature (+, +, −, −). Let us choose a basis for 𝕋 such that

ϕ = ( \begin{matrix}0 & iI_{2} \\ −iI_{2} & 0 \end{matrix} )

where I_{2} ∈ M_{2}(ℂ) denotes the identity matrix. For A ∈ M_{2}(ℂ), we have:

(A^{∗}, I_{2}) ϕ ( \begin{matrix}A \\ I_{2} \end{matrix} ) = 0 iff A is Hermitian.

In other words, the local chart

\begin{matrix}ℂ^{4} & ⟶ & 𝕄 \\ (y_{1}, y_{2}, y_{3}, y_{4}) & ↦ & ( \begin{matrix}y_{3} + y_{4} & y_{1} + iy_{2} \\ y_{1} − iy_{2} & y_{3} − y_{4} \\ 1 & 0 \\ 0 & 1 \end{matrix} ) \end{matrix}

identifies the Minkowski space M^{4} = (ℝ^{4}, ϕ) to an open subset of the completely real compact submanifold M of 𝕄 defined by:

M = {L_{2} ∈ 𝕄; ϕ(v) = 0 ∀v ∈ L_{2}}.

Note that M is a conformal compactification of the Minkowski space M^{4}. Let us consider

F = {(L_{1}, L_{2}) ∈ 𝔽; ϕ(v) = 0 ∀v ∈ L_{2}},
P = {L_{1} ∈ ℙ; ϕ(v) = 0 ∀v ∈ L_{1}},

and the induced correspondence

F_{N N N Q} _{f}  _{ȷȷȷ fi} _{g} P  M ↪ ℙ  𝕄.^{𝔽 N N N Q_{f}  _{ȷȷȷ fi}_{g}}

Recall that 𝕄 is a complexification of M, that P is a real hypersurface of ℙ topologically isomorphic to S^{2} × S^{3}, and that f̃ is locally isomorphic to a projection P × S^{1} ⟶ P (cf [28]).

The sheaves A_{M} and B_{M} of analytic functions and Sato hyperfunctions respectively are given by

A_{M} = ℂ_{M} ⊗ O_{𝕄},
B_{M} = RHom(D′_{𝕄}ℂ_{M}, O_{𝕄}).

In order to apply formulas (2.20) and (2.21), let us calculate Φ_{𝔽}G for G = ℂ_{M}[−2].

Since g^{−1}(M) = F, we have g^{−1}ℂ_{M} = ℂ_{F}. Moreover, since f̃ is locally isomorphic to P × S^{1} ⟶ P, we find that Φ_{𝔽}G = Rf_{∗}ℂ_{F} is concentrated in degree 0, 1 and H^{j}(Φ_{𝔽}G) is locally free of rank one for j = 0 or 1. Finally, since P ≃ S^{2} × S^{3} is connected and simply connected, we have H^{0}(Φ_{𝔽}G) ≃ ℂ_{P}, H^{1}(Φ_{𝔽}G) ≃ ℂ_{P}.

Hence, the distinguished triangle:

τ^{≤0}Φ_{𝔽}G ⟶ Φ_{𝔽}G ⟶ τ^{≥1}Φ_{𝔽}G ⟶_{+1}

is isomorphic to the distinguished triangle

ℂ_{P} ⟶ Φ_{𝔽}G ⟶ ℂ_{P} [−1] ⟶_{+1} . (5.2)

Applying the functor D′_{ℙ}(·) and using the isomorphism D′_{ℙ}(ℂ_{P}) ≃ or_{P} [−1] ≃ ℂ_{P} [−1], we get the distinguished triangle

ℂ_{P} ⟶ D′_{ℙ}Φ_{𝔽}G ⟶ ℂ_{P} [−1] ⟶_{+1} . (5.3)

Note that since H^{2}(P; ℂ_{P}) is different from 0, we do not know whether these triangles split or not (i.e. whether Φ_{𝔽}G ≃ ℂ_{P} ⊕ ℂ_{P} [−1]).

Now we assume k < 0 (5.4)

so that \underline{Φ}_{𝔽}D_{ℙ}(−k) is a coherent D_{𝕄}-module (concentrated in degree zero) by Proposition 5.2.

Applying the functors RΓ(ℙ; · ⊗ O_{ℙ}(k)) and RΓ(ℙ; RHom(·, O_{ℙ}(k))) to (5.2) and (5.3) respectively, we get the long exact sequences:

\begin{aligned}⋯ & ⟶ & H^{j+1}(P; O_{ℙ}(k)) ⟶ H^{j+1}RΓ(ℙ; Φ_{𝔽}G ⊗ O_{ℙ}(k)) ⟶ \\ & ⟶ & H^{j}(P; O_{ℙ}(k)) ⟶ H^{j+2}(P; O_{ℙ}(k)) ⟶ ⋯ \end{aligned} (5.5)

and

\begin{aligned}⋯ & ⟶ & H_{P}^{j+1}(ℙ; O_{ℙ}(k)) ⟶ H^{j}RΓ(ℙ; RHom(D′_{ℙ}Φ_{𝔽}G, O_{ℙ}(k))) ⟶ \\ & ⟶ & H_{P}^{j} (ℙ; O_{ℙ}(k)) ⟶ H_{P}^{j+2}(ℙ; O_{ℙ}(k)) ⟶ ⋯ \end{aligned} (5.6)

Taking the cohomology of degree −3 in (2.20), (2.21), we get:

H^{0}(RΓ(ℙ; Φ_{𝔽}G ⊗ O_{ℙ}(k))) ≃ H^{−1}RΓ(𝕄; RHom_{D_{𝕄}}(\underline{Φ}_{𝔽}D_{ℙ}(−k), A_{M})),

H^{0}(RΓ(ℙ; RHom(D′Φ_{𝔽}G, O_{ℙ}(k)))) ≃ H^{−1}RΓ(𝕄; RHom_{D_{𝕄}}(\underline{Φ}_{𝔽}D_{ℙ}(−k), B_{M})),

and the terms on the right hand side both vanish by hypothesis (5.4) and Proposition 5.2. Hence we get from (5.5) and (5.6)

\{ \begin{matrix}H^{0}(P; O_{ℙ}(k)) = 0 \\ H_{P}^{1} (ℙ; O_{ℙ}(k)) = 0 \end{matrix}

Taking the zero-th cohomology group in (2.20), (2.21), we find the isomorphisms:

H^{1}(P; O_{ℙ}(k)) ≃ Hom_{D_{𝕄}}(\underline{Φ}_{𝔽}D_{ℙ}(−k), A_{M}),
H_{P}^{2} (ℙ; O_{ℙ}(k)) ≃ Hom_{D_{𝕄}}(\underline{Φ}_{𝔽}D_{ℙ}(−k), B_{M}).

This is Theorem 6.1 of [29].

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