Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I

TL;DR

This paper develops a gluing theory for pseudo-holomorphic curves in symplectic geometry, enabling the calculation of how Gromov-Witten invariants of Calabi-Yau 3-folds change under flops and extremal transitions.

Contribution

It introduces a general gluing framework for relative Gromov-Witten invariants and provides explicit formulas for invariants' transformations under specific geometric transitions.

Findings

Complete formula for GW-invariants change under flop

Formula for GW-invariants change under type I extremal transition

Framework applicable to symplectic cutting and contact surgery

Abstract

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.