Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The p-cycle of Holonomic D-modules and Quantization of Exact Algebraic Lagrangians

Christopher Dodd

Abstract

Let $X=\mathbb{A}^{n}$ be complex affine space, and let $T^{*}X$ be its cotangent bundle. For any exact Lagrangian $L\subset T^{*}X$, we define a new invariant, A, living in $ \text{Div}_{\mathbb{Q}/\mathbb{Z}}(L)$. We call this invariant the monodromy divisor of $L$. We conjecture that the existence of a finite order character of $π_{1}(L$) whose monodromy is exactly A defines an obstruction to attaching a holonomic $\mathcal{D}_{X}$-module M associated to L - here, the association goes via positive characteristic and p-supports. In the case where $\mathbb{H}_{dR}^{1}(L)=0$, we prove this conjecture, and then go on the show that the set of such holonomic $\mathcal{D}_{X}$-modules forms a torsor over the group of finite order characters of $π_{1}$. This proves a version of a conjecture of Kontsevich. As a consequence, we deduce that the group of Morita autoequivalences of the n-th Weyl algebra is isomorphic to the group of symplectomorphisms of $T^{*}\mathbb{A}^{n}$. This generalizes an old theorem of Dixmier (in the case n=1) and settles a conjecture of Belov-Kanel and Kontsevich in general.

The p-cycle of Holonomic D-modules and Quantization of Exact Algebraic Lagrangians

Abstract

Let be complex affine space, and let be its cotangent bundle. For any exact Lagrangian , we define a new invariant, A, living in . We call this invariant the monodromy divisor of . We conjecture that the existence of a finite order character of ) whose monodromy is exactly A defines an obstruction to attaching a holonomic -module M associated to L - here, the association goes via positive characteristic and p-supports. In the case where , we prove this conjecture, and then go on the show that the set of such holonomic -modules forms a torsor over the group of finite order characters of . This proves a version of a conjecture of Kontsevich. As a consequence, we deduce that the group of Morita autoequivalences of the n-th Weyl algebra is isomorphic to the group of symplectomorphisms of . This generalizes an old theorem of Dixmier (in the case n=1) and settles a conjecture of Belov-Kanel and Kontsevich in general.

Paper Structure

This paper contains 33 sections, 100 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Key ideas of the paper
  3. Notations and Conventions
  4. Acknowledgements
  5. The Monodromy Divisor of an Exact Lagrangian
  6. The Generic Structure of an exact Lagrangian
  7. The monodromy divisor of an exact Lagrangian
  8. Invariance
  9. Meromorphic Connections
  10. Higgs Sheaves
  11. $\theta$-regular $\lambda$-connections in positive characteristic
  12. Infinitesimal $\theta$-regular Connections
  13. Local Deformation theory
  14. Construction of $\theta$-regular sheaves over $R[\lambda]/\lambda^{2}$
  15. $\theta$-regular connections over $R[\lambda]/\lambda^{n}$
  16. ...and 18 more sections

Key Result

Theorem 1.3

Let $L_{\mathbb{C}}=X_{\mathbb{C}}\subset T^{*}X_{\mathbb{C}}$. Then a holonomic $\mathcal{D}_{X_{\mathbb{C}}}$-module $\mathcal{M}_{\mathbb{C}}$ has constant arithmetic support equal to $L_{\mathbb{C}}$, with multiplicity $1$, iff $\mathcal{M}_{\mathbb{C}}$ is a line bundle with flat connection, wh

Theorems & Definitions (207)

  • Definition 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Proposition 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • proof
  • ...and 197 more