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Lagrangian structures on the derived moduli of constructible sheaves

Merlin Christ, Enrico Lampetti

Abstract

Given a compact oriented manifold of dimension $n$ with a conically smooth stratification, we show that the moduli of $\mathcal{D}(k)$-valued constructible sheaves and the moduli of perverse sheaves are $(2-n)$-shifted Lagrangian. The former statement follows from the construction of a relative left $n$-Calabi--Yau structure on the stable $\infty$-category of $\mathcal{D}(k)$-valued constructible sheaves. This is achieved via a lax gluing result for categorical cubes equipped with cubical Calabi--Yau structures. Given a codimension $2$ submanifold, we further identify symplectic leaves corresponding to perverse sheaves with prescribed monodromy.

Lagrangian structures on the derived moduli of constructible sheaves

Abstract

Given a compact oriented manifold of dimension with a conically smooth stratification, we show that the moduli of -valued constructible sheaves and the moduli of perverse sheaves are -shifted Lagrangian. The former statement follows from the construction of a relative left -Calabi--Yau structure on the stable -category of -valued constructible sheaves. This is achieved via a lax gluing result for categorical cubes equipped with cubical Calabi--Yau structures. Given a codimension submanifold, we further identify symplectic leaves corresponding to perverse sheaves with prescribed monodromy.
Paper Structure (19 sections, 43 theorems, 135 equations)

This paper contains 19 sections, 43 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Calabi--Yau categorical cubes
  3. Linear stable infinity-categories and categorical cubes
  4. Cubical Calabi--Yau structures
  5. Lax gluing
  6. Non-lax gluing
  7. Calabi--Yau structures on constructible sheaves categories
  8. Constructible sheaves on stratified spaces
  9. Conically smooth stratified spaces
  10. The categorical cube of a conically smooth stratified space
  11. Calabi--Yau cubes associated with smooth manifolds with corners
  12. The cubical Calabi--Yau structure on Cons(X)
  13. An illustrated example
  14. Lagrangian structures on moduli of constructible sheaves
  15. Moduli of objects
  16. ...and 4 more sections

Key Result

Theorem 1

Let $X$ be a compact oriented manifold of dimension $d$ with boundary. Let $(X,P)$ be a conically smooth stratification of $X$ with boundary of depth $n$.

Theorems & Definitions (132)

  • Theorem 1: \ref{['thm:final_result']}, \ref{['corollary:lagrangian_general']}
  • Remark 1
  • Corollary 1: \ref{['coro:Poisson_structure']}
  • Example 1
  • Theorem 2: \ref{['thm:symplectic_leaves']}
  • Remark 2
  • Example 2
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 122 more