A natural Gromov-Witten virtual fundamental class

TL;DR

This paper presents a method to construct a unique virtual fundamental class for Gromov-Witten moduli spaces of compact symplectic manifolds, ensuring naturality without relying on gluing theorems in certain cases.

Contribution

It introduces a novel approach using Ruan-Tian perturbations, stabilizing divisors, and Cech homology to establish the virtual fundamental class with naturality conditions.

Findings

Constructs a unique virtual fundamental class for Gromov-Witten moduli spaces.

Eliminates the need for gluing theorems in high dimensions or low genus.

Ensures the class satisfies naturality conditions.

Abstract

We describe a program for proving that the Gromov-Witten moduli spaces of compact symplectic manifolds carry a unique virtual fundamental class that satisfies certain naturality conditions. The virtual fundamental class is constructed using only Ruan-Tian perturbations by introducing stabilizing divisors, using Cech homology, and systematically applying naturality conditions. In high dimensions or low genus, no gluing theorems are needed.