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Relative orbifold Gromov-Witten theory and degeneration formula

Bohui Chen, An-Min Li, Shanzhong Sun, Guosong Zhao

TL;DR

This work develops a comprehensive framework for relative Gromov-Witten theory in the orbifold setting by integrating the Li–Ruan degeneration approach with Chen-Ruan orbifold GW theory. It constructs moduli spaces of relative orbifold stable maps, proves their compactness, and equips them with virtual fundamental cycles via a Kuranishi-structure approach, including top and lower strata and gluing analysis. The central result is a degeneration formula that expresses orbifold GW invariants of a degenerated target as sums of relative invariants on the split pieces, incorporating fractional contact data and orbifold monodromies. The methodology advances the understanding of how orbifold structure interacts with degeneration and sets the stage for applications to invariance questions in orbifold quantum cohomology under orbifold flops in dimension three.

Abstract

Relative orbifold Gromov-Witten theory is set-up and the degeneration formula is given.

Relative orbifold Gromov-Witten theory and degeneration formula

TL;DR

This work develops a comprehensive framework for relative Gromov-Witten theory in the orbifold setting by integrating the Li–Ruan degeneration approach with Chen-Ruan orbifold GW theory. It constructs moduli spaces of relative orbifold stable maps, proves their compactness, and equips them with virtual fundamental cycles via a Kuranishi-structure approach, including top and lower strata and gluing analysis. The central result is a degeneration formula that expresses orbifold GW invariants of a degenerated target as sums of relative invariants on the split pieces, incorporating fractional contact data and orbifold monodromies. The methodology advances the understanding of how orbifold structure interacts with degeneration and sets the stage for applications to invariance questions in orbifold quantum cohomology under orbifold flops in dimension three.

Abstract

Relative orbifold Gromov-Witten theory is set-up and the degeneration formula is given.

Paper Structure

This paper contains 53 sections, 23 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Review of relative Gromov-Witten theory
  3. Relative pairs and degeneration
  4. Neighborhood of divisor
  5. Symplectic manifold with cylindric ends
  6. Degeneration
  7. Degeneration of $X$ along $Z$
  8. Moduli space of relative stable maps
  9. Relative stable maps
  10. Dual graph and stratification
  11. Review of orbifold Gromov-Witten theory
  12. Basic orbifold theory
  13. Orbifold structure
  14. Orbifold bundles
  15. Orbifold morphisms
  16. ...and 38 more sections

Key Result

Theorem 1.1

$\overline{\mathcal{M}}_{g,(\mathbf g),A,(\mathbf h),T_k}(\mathsf G,\mathsf Z)$ is compact and carries a virtual fundamental cycle.

Theorems & Definitions (51)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 41 more