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On Lagrangianity of $p$-supports of holonomic D-modules and $q$-D-modules

Pavel Etingof

Abstract

M. Kontsevich conjectured and T. Bitoun proved that if M is a nonzero holonomic D-module then the p-support of a generic reduction of M to characteristic p>0 is Lagrangian. We provide a new elementary proof of this theorem and also generalize it to q-D-modules. The proofs are based on Bernstein's theorem that any holonomic D-module can be transformed by an element of the symplectic group into a vector bundle with a flat connection, and a q-analog of this theorem. We also discuss potential applications to quantizations of symplectic singularities and to quantum cluster algebras.

On Lagrangianity of $p$-supports of holonomic D-modules and $q$-D-modules

Abstract

M. Kontsevich conjectured and T. Bitoun proved that if M is a nonzero holonomic D-module then the p-support of a generic reduction of M to characteristic p>0 is Lagrangian. We provide a new elementary proof of this theorem and also generalize it to q-D-modules. The proofs are based on Bernstein's theorem that any holonomic D-module can be transformed by an element of the symplectic group into a vector bundle with a flat connection, and a q-analog of this theorem. We also discuss potential applications to quantizations of symplectic singularities and to quantum cluster algebras.
Paper Structure (17 sections, 19 theorems, 49 equations)

This paper contains 17 sections, 19 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Lagrangianity of $p$-supports of a holonomic D-module
  3. Flat connections and $p$-curvature
  4. Lagrangian flat connections in characteristic $p$
  5. Liftable connections
  6. Bernstein's theorem
  7. Bitoun's theorem
  8. $p$-supports of holonomic modules over quantized conical symplectic singularities
  9. Lagrangianity of $p$-supports of a holonomic $q$-D-module
  10. Flat $q$-connections and $q$-D-modules
  11. Freeness of $\mathcal{O}$-coherent $q$-D-modules
  12. $q$-analog of Bernstein's theorem
  13. $N$-curvature
  14. Lagrangian flat $q$-connections
  15. Liftable $q$-connections
  16. ...and 2 more sections

Key Result

Proposition 2.1

$\nabla$ is Lagrangian if and only if ${\rm supp}\nabla$ is a Lagrangian subvariety of $T^*(\Bbb A^r)^{(1)}$.

Theorems & Definitions (43)

  • Proposition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • ...and 33 more