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Properties of Gromov-Witten invariants defined via global Kuranishi charts

Amanda Hirschi

Abstract

Using the global Kuranishi charts constructed in \cite{HS22}, we define gravitational descendants and equivariant Gromov-Witten invariants for general symplectic manifolds. We prove that that these invariants, equivariant and non-equivariant, satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it derive an explicit formula for the equivariant GW invariants of a class of Hamiltonian manifolds. A comparison with the GW invariants of \cite{RT97} is given in the semipositive case.

Properties of Gromov-Witten invariants defined via global Kuranishi charts

Abstract

Using the global Kuranishi charts constructed in \cite{HS22}, we define gravitational descendants and equivariant Gromov-Witten invariants for general symplectic manifolds. We prove that that these invariants, equivariant and non-equivariant, satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it derive an explicit formula for the equivariant GW invariants of a class of Hamiltonian manifolds. A comparison with the GW invariants of \cite{RT97} is given in the semipositive case.
Paper Structure (27 sections, 65 theorems, 224 equations)

This paper contains 27 sections, 65 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Global Kuranishi charts
  4. Main results
  5. Review of the global Kuranishi chart construction
  6. Kontsevich--Manin axioms
  7. The first five axioms
  8. Fundamental class axiom
  9. Divisor axiom
  10. Splitting axiom
  11. Genus reduction axiom
  12. Gravitational descendants
  13. Virtual localisation and equivariant GW theory
  14. Equivariant virtual fundamental classes
  15. Virtual localisation
  16. ...and 12 more sections

Key Result

Theorem 1.2

The Gromov--Witten classes of $(X,\omega)$ satisfy the Kontsevich-Manin axioms, listed below.

Theorems & Definitions (160)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 1.4: Theorem \ref{['thm:formula-eq-gw']}
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7: Theorem \ref{['pseudocycle-comparison']}
  • Corollary 1.8
  • Definition 2.1
  • Remark 2.3
  • ...and 150 more