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Gromov--Witten theory beyond maximal contacts

Yu Wang, Fenglong You

Abstract

Given a smooth projective variety $X$ and a smooth nef divisor $D$, we identify genus zero relative Gromov--Witten invariants of $(X,D)$ with $(n+1)$ relative markings with genus zero relative/orbifold Gromov--Witten invariants of a $\mathbb P^1$-bundle $P:=\mathbb P(\mathcal O_X(-D)\oplus \mathcal O_X)$ with $n$ relative markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.

Gromov--Witten theory beyond maximal contacts

Abstract

Given a smooth projective variety and a smooth nef divisor , we identify genus zero relative Gromov--Witten invariants of with relative markings with genus zero relative/orbifold Gromov--Witten invariants of a -bundle with relative markings. This is a generalization of the local-relative correspondence beyond maximal contacts. Repeating this process, we identify genus zero relative Gromov--Witten invariants with genus zero absolute Gromov--Witten invariants of toric bundles. We also present how this correspondence can be used to compute genus zero two-point relative Gromov--Witten invariants.
Paper Structure (25 sections, 15 theorems, 88 equations)

This paper contains 25 sections, 15 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. The degeneration scheme
  3. The local-relative correspondence
  4. Main theorems
  5. Computation of genus zero relative Gromov--Witten invariants
  6. Future directions
  7. The degeneration formula for orbifolds
  8. Proof of the main theorem
  9. Degeneration
  10. Virtual cycles in the degeneration formula
  11. Virtual cycles on the $(X\times \mathbb P^1)$-side
  12. Virtual cycles on the $P_Y$-side
  13. The degeneration graphs
  14. Edges
  15. Vertices with $p_i$ attached.
  16. ...and 10 more sections

Key Result

Theorem 1.3

Given a smooth projective variety $X$ with a smooth nef divisor $D$, genus zero relative invariants of $(X,D)$ satisfy the following identity: where the RHS are the virtual cycles for the moduli spaces of genus zero orbifold stable maps to the pair $(P,X_{\infty}+ X_{\sigma})$ with contact orders as defined in TY20c.

Theorems & Definitions (33)

  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Corollary 1.10
  • Theorem 1.11: =You22*Theorem 5.1
  • Remark 1.12
  • ...and 23 more