Micrologicalization of perverse sheaves

Andronikov, E.

doi:10.1007/bf02362631

Cited by 3 publications

(1 citation statement)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.

View full text Add to dashboard Cite

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...

show abstract

Section: Introductionmentioning

confidence: 99%

The codimension-three conjecture for holonomic DQ-modules

Petit

2017

Sel. Math. New Ser.

View full text Add to dashboard Cite

show abstract

Bibliography

2006

An Introduction to Intersection Homology Theory, Second Edition

View full text Add to dashboard Cite

Microdifferential systems and the codimension-three conjecture

Kashiwara

Vilonen

2014

Ann. Math.

View full text Add to dashboard Cite

Introduction 1 2. Microdifferential operators 7 3. Construction of commutative rings 11 4. Reduction via finite morphisms 13 5. Proof of the main theorems 16 5.1. General preliminaries 16 5.2. The formal codimension three extension theorem 17 5.3. The formal codimension two submodule extension theorem 18 5.4. The codimension three extension theorem 19 5.5. The codimension two submodule extension theorem 20 6. The commutative formal codimension three extension theorem 20 7. The commutative formal submodule extension theorem 32 8. Comparison of the formal and convergent cases 35 9. Open problems 45 References 46

show abstract

Micrologicalization of perverse sheaves

Cited by 3 publications

References 10 publications

The codimension-three conjecture for holonomic DQ-modules

The codimension-three conjecture for holonomic DQ-modules

Bibliography

Microdifferential systems and the codimension-three conjecture

Contact Info

Product

Resources

About