A microlocal version of the Riemann-Hilbert correspondence

Andronikof, Emmanuel

doi:10.12775/tmna.1994.036

Cited by 10 publications

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“…-Let R + denote the group of strictly positive real numbers and C × the group of non-zero complex numbers. We will mainly work on a fixed complex manifold (1) X of complex dimension dim C X = d X . Let T its cotangent bundle and T * X X the zero section.…”

Section: Microlocalization Of Sheavesmentioning

confidence: 99%

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The stack of microlocal perverse sheaves

Waschkies¹

2004

Bul. Soc. Math. France

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Abstract. -In this paper we construct the abelian stack of microlocal perverse sheaves on the projective cotangent bundle of a complex manifold. Following ideas of Andronikof we first consider microlocal perverse sheaves at a point using classical tools from microlocal sheaf theory. Then we will use Kashiwara-Schapira's theory of analytic ind-sheaves to globalize our construction. This presentation allows us to formulate explicitly a global microlocal Riemann-Hilbert correspondence.Résumé (Le champ des faisceaux pervers microlocaux ). -Nous construisons le champ abélien des faisceaux pervers microlocaux sur le fibré cotangent projectif d'une variété analytique complexe. Suivant des idées d'Andronikof, nous considérons d'abord les germes de faisceaux pervers microlocaux en un point en utilisant les outils classiques de la théorie microlocale des faisceaux. Ensuite nous utilisons la théorie des ind-faisceaux analytiques de Kashiwara-Schapira pour globaliser notre construction. Cette présentation nous permettra de formuler explicitement une version globale de la correspondance de Riemann-Hilbert microlocale.

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Section: Microlocalization Of Sheavesmentioning

confidence: 99%

“…In [1] Andronikof defined a prestack on P * X and announced the microlocal Riemann-Hilbert correspondence on the stalks. However, at that time there did not exist tools to define a global microlocal Riemann-Hilbert morphism.…”

Section: Introductionmentioning

confidence: 99%

The stack of microlocal perverse sheaves

Waschkies¹

2004

Bul. Soc. Math. France

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“…One of the important aspects of the codimension-three conjecture for microdifferential modules is that it provides essential information on the structure of the stack of microlocal perverse sheaves through the microlocal Riemann-Hilbert correspondence (see [1], [2], [23]). However, the interpretation of the codimension-three conjecture for holonomic third section, we adapt the coherence criterion from [18] corresponding to Proposition 4.1 and 4.4 of [14] to the case of DQ-modules.…”

Section: Introductionmentioning

confidence: 99%

The codimension-three conjecture for holonomic DQ-modules

Petit

2017

Sel. Math. New Ser.

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We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely beyond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module.The author was supported by the EPSRC grant EP/G007632/1. 1 If codim S ≥ 3, is it true that any locally free (or even only reflexive) sheaf on X − S is extendable ?bundle of a complex manifold. It was recently proved by Kashiwara and Vilonen (see [14]). Theorem 1.2 ([14, Theorem 1.2]). Let X be a complex manifold, U an open subset of T * X, Λ a closed Lagrangian analytic subset of U, and Y a closed analytic subset of Λ such that codim Λ Y ≥ 3. Let E X the sheaf of microdifferential operators on T * X and M be a holonomic (E X | U \Y )-module whose support is contained in Λ \ Y . Assume that M possesses an (E X (0)| U \Y )-lattice. Then M extends uniquely to a holonomic module defined on U whose support is contained in Λ.The proof of the conjecture was made possible by the deep result of Kashiwara and Vilonen extending Theorem 1.1 to coherent sheavesDQ-modules in terms of perverse sheaves is not yet clear since the analogue of microlocal perverse sheaves is not known for DQ-modules. It is an interesting problem to define such objects.Here are the precise statements of the results we are proving.Theorem 1.4. Let X be a complex manifold endowed with a DQalgebroid stack A X such that the associated Poisson structure is symplectic. Let Λ be a closed Lagrangian analytic subset of X and Y a closed analytic subset of Λ such that codim Λ Y ≥ 3. Let M be a holonomic (A loc X | X\Y )-module, whose support is contained in Λ \ Y . Assume that M has an A X | X\Y -lattice. Then M extends uniquely to a holonomic module defined on X whose support is contained in Λ. and Theorem 1.5. Let X be a complex manifold endowed with a DQalgebroid stack A X such that the associated Poisson structure is symplectic. Let Λ be a closed Lagrangian analytic subset of X and Y a closed analytic subset of Λ such that codim Λ Y ≥ 2. Let M be a holonomic A loc X -module whose support is contained in Λ and let M 1 be an A loc X | X\Y -submodule of M| X\Y . Then M 1 extends uniquely to a holonomic A loc X -submodule of M. We hope that the submodule version of the codimension-three conjecture for DQ-modules may have applications to the representation theory of Cherednik algebras, via the localization theorem due to Kashiwara and Rouquier (see [10]).Let us describe briefly the proof of the above statements. As we already remarked, we follow the general strategy of Kashiwara and Vilonen (see [14]). Keeping the notations of the theorems 1.4 and 1.5 and denoting by j : X \ Y → X the open embedding of X \ Y into X, the problem essentially amount to show that j * M and j * M 1 are coherent. This is a local question. By adapting a standard technique in several complex variables in [18] to the cas...

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“…Now let us formulate the comparison theorem for regular holonomic E Xmodules in terms of ind-sheaves. The classical version ([An1], Proposition 5.6.3) states Proposition 3.5.3.…”

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confidence: 99%

Microlocal Riemann–Hilbert Correspondence

Waschkies¹

2005

Publ. Res. Inst. Math. Sci.

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We construct the global microlocal Riemann-Hilbert correspondence as an explicit equivalence between the abelian stack of microlocal perverse sheaves defined in [W] and the abelian stack of regular holonomic microdifferential modules of [KK]. The theory of analytic ind-sheaves and its microlocalization is crucial for our construction since it allows us to define solution complexes with values in the (ind-)ring of microlocal holomorphic functions (resp. microlocal tempered holomorphic functions).

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A microlocal version of the Riemann-Hilbert correspondence

Cited by 10 publications

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The stack of microlocal perverse sheaves

The stack of microlocal perverse sheaves

The codimension-three conjecture for holonomic DQ-modules

Microlocal Riemann–Hilbert Correspondence

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