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Cosection localization via shifted symplectic geometry

Young-Hoon Kiem, Hyeonjun Park

Abstract

The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived Deligne-Mumford stack with a $(-1)$-shifted closed $1$-form is equal to the virtual Lagrangian cycle of the degeneracy locus which is $(-2)$-shifted symplectic. We next establish a shifted analogue of the Lagrange multipliers method which gives us the quantum Lefschetz theorems as immediate consequences of the equality of virtual cycles. Lastly we study derived algebraic geometry enhancements of gauged linear sigma models which lead us to the relative virtual cycles in a general and natural form.

Cosection localization via shifted symplectic geometry

Abstract

The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic geometry. We first prove that the cosection localized virtual cycle of a quasi-smooth derived Deligne-Mumford stack with a -shifted closed -form is equal to the virtual Lagrangian cycle of the degeneracy locus which is -shifted symplectic. We next establish a shifted analogue of the Lagrange multipliers method which gives us the quantum Lefschetz theorems as immediate consequences of the equality of virtual cycles. Lastly we study derived algebraic geometry enhancements of gauged linear sigma models which lead us to the relative virtual cycles in a general and natural form.
Paper Structure (48 sections, 43 theorems, 236 equations)

This paper contains 48 sections, 43 theorems, 236 equations.

Table of Contents

  1. Virtual fundamental class
  2. Cosection localization
  3. Shifted symplectic structure on the degeneracy locus of a closed cosection
  4. Comparison of virtual cycles
  5. Shifted Lagrange multipliers method and quantum Lefschetz
  6. Gauged linear sigma models
  7. The layout
  8. Cosection localization revisited
  9. Virtual fundamental class
  10. Virtual pullback
  11. Cosection localized virtual fundamental class
  12. Cosection localized virtual pullback
  13. Locked cosections and localized virtual pullback
  14. Cosection localization, derived
  15. Localized Gysin maps for vector bundles
  16. ...and 33 more sections

Key Result

Theorem A

Let $M$ be a quasi-smooth derived Deligne-Mumford stack and $\boldsymbol\sigma$ be a $(-1)$-shifted closed $1$-form on $M$. Denote by $M(\sigma)$ the degeneracy locus. Then we have the equality of the cosection localized virtual cycle i2 and the virtual Lagrangian cycle i4.

Theorems & Definitions (100)

  • Theorem A: Theorem \ref{['Thm:main']}
  • Theorem B: Theorem \ref{['Prop:LagMult']}
  • Corollary C: Theorem \ref{['Cor:Lefschetz']}
  • Theorem D: Theorem \ref{['Prop:GLSMmain']}
  • Corollary E: Corollary \ref{['Cor:2']}
  • Definition 1.1
  • Lemma 1.2
  • Proposition 1.3
  • Lemma 1.6
  • Example 1.7
  • ...and 90 more