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Moduli of sheaves on fourfolds as derived Lagrangian intersections

Nachiketa Adhikari, Yun Shi

Abstract

We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any $n$-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. We also give a separate proof of our main result in the case where the local Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree $(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$.

Moduli of sheaves on fourfolds as derived Lagrangian intersections

Abstract

We show that any -shifted symplectic derived scheme (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a -shifted symplectic derived scheme which is the -shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any -shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. We also give a separate proof of our main result in the case where the local Darboux atlas cdga for has an even number of generators in degree ; in this case we strengthen the result by showing that is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is .
Paper Structure (13 sections, 23 theorems, 84 equations)

This paper contains 13 sections, 23 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Generalizations
  3. Notation
  4. Acknowledgments
  5. Realization as a Lagrangian intersection
  6. The even case
  7. The general case
  8. The derived moduli stack of coherent sheaves
  9. Resolutions of the polynomial algebra
  10. Construction
  11. Universal object
  12. Pullback of the symplectic structure to an atlas
  13. The moduli of sheaves on four-dimensional affine space as a Lagrangian intersection

Key Result

Theorem 1.1

Let $\bf F$ be an $n$-shifted symplectic derived Artin stack and ${\bf X} \to {\bf F}$, ${\bf Y} \to {\bf F}$ two morphisms of derived Artin stacks carrying Lagrangian structures. Then the derived stack ${\bf X} \times^h_{\bf F} {\bf Y}$ has a canonical $(n-1)$-shifted symplectic structure.

Theorems & Definitions (39)

  • Theorem 1.1: ptvv
  • Theorem 1.2: ptvv, bd19
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Proposition 2.5
  • ...and 29 more