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A categorification of the quantum Lefschetz principle

David Kern

Abstract

The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus $0$, regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the $\infty$-categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to recover the classical Quantum Lefschetz formula when its hypotheses are satisfied.

A categorification of the quantum Lefschetz principle

Abstract

The quantum Lefschetz formula explains how virtual fundamental classes (or structure sheaves) of moduli stacks of stable maps behave when passing from an ambient target scheme to the zero locus of a section. It is only valid under special assumptions (genus , regularity of the section and convexity of the bundle). In this paper, we give a general statement at the geometric level removing these assumptions, using derived geometry. Through a study of the structure sheaves of derived zero loci we deduce a categorification of the formula in the -categories of quasi-coherent sheaves. We also prove that Manolache's virtual pullbacks can be constructed as derived pullbacks, and use them to recover the classical Quantum Lefschetz formula when its hypotheses are satisfied.

Paper Structure

This paper contains 16 sections, 18 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The quantum Lefschetz hyperplane principle
  3. Derived moduli stacks and virtual classes
  4. Acknowledgements
  5. Notations and conventions
  6. Zero loci of sections of derived vector bundles
  7. A spicilege of derived geometry for Gromov--Witten theory
  8. Vector bundles in derived geometry
  9. Excess intersection formula
  10. The geometric Lefschetz principle
  11. Review of the derived moduli stack of stable maps
  12. Identification of the derived moduli stacks
  13. Functoriality in intersection theory by the categorification of virtual pullbacks
  14. Definition from derived geometry
  15. Comparison with the construction from obstruction theories
  16. ...and 1 more sections

Key Result

Theorem A

For any $\gamma\in A_{1}Z$ such that $i_{\ast}\gamma=\beta$, let $u_{\gamma}\colon\overline{\mathcal{M}}_{0,n}(Z,\gamma) \hookrightarrow\overline{\mathcal{M}}_{0,n}(X,\beta)$ denote the closed immersion. Suppose $E$ is convex, that is $\mathbb{R}^{1}p_{\ast}(C,\mu^{\ast}E)=0$ for any stable map $\mu and

Theorems & Definitions (57)

  • Theorem A: kim03:_funct_cox_katz_leejoshua10:_rieman
  • Theorem B: Categorified quantum Lefschetz principle, see \ref{['corlr:main-result']} and \ref{['pros:zerosec-struct-shf']}
  • Remark C
  • Remark D
  • Example 2.0.1: Cotangent complex
  • Example 2.0.2: Quasicoherent modules
  • Definition 2.1.1: Total space of a quasicoherent module
  • Remark 2.1.2
  • Remark 2.1.3
  • Example 2.1.5
  • ...and 47 more