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Splitting of Gromov-Witten Invariants with Toric Gluing Strata

Yixian Wu

Abstract

We prove a splitting formula that reconstructs the logarithmic Gromov- Witten invariants of simple normal crossing varieties from the punctured Gromov- Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov-Witten theory developed in arXiv:2009.07720.

Splitting of Gromov-Witten Invariants with Toric Gluing Strata

Abstract

We prove a splitting formula that reconstructs the logarithmic Gromov- Witten invariants of simple normal crossing varieties from the punctured Gromov- Witten invariants of their irreducible components, under the assumption of the gluing strata being toric varieties. The formula is based on the punctured Gromov-Witten theory developed in arXiv:2009.07720.

Paper Structure

This paper contains 17 sections, 30 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Idea of the proof and structure of the paper
  4. Other approaches
  5. Acknowledgement
  6. Conventions
  7. Punctured invariants and the gluing formalism
  8. Punctured Gromov-Witten invariants
  9. Logarithmic evaluation maps
  10. The gluing formalism
  11. Generalization of Fulton-Sturmfels formula
  12. Proof of the splitting formula
  13. Toric Strata Assumption
  14. Local Toric Models of the Gluing Formalism
  15. Gluing of the Local Models
  16. ...and 2 more sections

Key Result

Theorem 1.1

ACGSPunc There is a finite, representable morphism of moduli spaces of punctured log stable maps to $X$ over $B$

Theorems & Definitions (65)

  • Theorem 1.1
  • Lemma 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Corollary 1.7
  • proof
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • ...and 55 more