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Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties

Gwyn Bellamy, Christopher Dodd, Kevin McGerty, Thomas Nevins

Abstract

We prove a new symplectic analogue of Kashiwara's Equivalence from D-module theory. As a consequence, we establish a structure theory for module categories over deformation quantizations that mirrors, at a higher categorical level, the Bialynicki-Birula stratification of a variety with an action of the multiplicative group. The resulting categorical cell decomposition provides an algebro-geometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K-theory and Hochschild homology of module categories of interest in geometric representation theory.

Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties

Abstract

We prove a new symplectic analogue of Kashiwara's Equivalence from D-module theory. As a consequence, we establish a structure theory for module categories over deformation quantizations that mirrors, at a higher categorical level, the Bialynicki-Birula stratification of a variety with an action of the multiplicative group. The resulting categorical cell decomposition provides an algebro-geometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K-theory and Hochschild homology of module categories of interest in geometric representation theory.

Paper Structure

This paper contains 50 sections, 104 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Symplectic Varieties with Elliptic ${\mathbb G}_m$-Action
  3. Categorical Cell Decomposition
  4. Analogue of Kashiwara's Equivalence for Deformation Quantizations
  5. Abelian and Derived Categories
  6. Abelian Categories
  7. Derived Categories and Compact Generation
  8. Applications to Invariants
  9. Relation to Fukaya Categories of Weinstein Manifolds
  10. Relation to other work on Deformation Quantization
  11. Outline of the Paper
  12. Conventions
  13. The Geometry of Symplectic Varieties with Elliptic ${\mathbb G}_m$-Action
  14. A Symplectic Białynicki-Birula Decomposition
  15. Proof of Theorem \ref{['thm:maincoiso']}(1)
  16. ...and 35 more sections

Key Result

Theorem 1.2

Theorems & Definitions (210)

  • Definition 1.1
  • Theorem 1.2: Theorem \ref{['thm:maincoiso']}
  • Theorem 1.3: Theorem \ref{['cor:sympafffactor']}
  • Definition 1.4
  • Theorem 1.5: Theorem \ref{['thm:equivmain1']} and Corollary \ref{['cor:filter']}
  • Corollary 1.6
  • Theorem 1.7: Theorem \ref{['thm:equivmain1']} and Corollary \ref{['cor:preshol']}
  • Theorem 1.8: Theorem \ref{['thm:equivmain1']} and Proposition \ref{['prop:twistedequiv']}
  • Theorem 1.9: Theorem \ref{['keyprop']} and Corollary \ref{['cor:qcohfullesssurj']}
  • Theorem 1.10: Corollaries \ref{['cor:compactgen']} and \ref{['cor:perfcompact']}
  • ...and 200 more