A Comparison Theorem for Gromov-Witten Invariants in the Symplectic Category

Abstract

In this paper we exploit the geometric approach to the virtual fundamental class, due to Fukaya-Ono and Li-Tian, to compare the virtual fundamental classes of stable maps to a symplectic manifold and a symplectic submanifold whenever all constrained stable maps to the former are contained in the latter to first order. This extends versions of a statement well-known in the algebraic category to the symplectic category, where it appears to be less familiar. The latter's inherent flexibility then leads to a confirmation of Pandharipande's Gopakumar-Vafa prediction for GW-invariants of Fano classes in 6-dimensional symplectic manifolds. In a forthcoming paper, we use a similar approach to relative Gromov-Witten invariants and the absolute/relative correspondence in genus~0.