Pseudo-holomorphic curves with a fixed complex structure in positive symplectic manifolds

TL;DR

This paper proves that in positive symplectic manifolds, fixed-domain Gromov-Witten invariants can be interpreted as signed counts of pseudo-holomorphic curves when using a generic almost complex structure, confirming a symplectic analogue of a conjecture.

Contribution

It establishes the validity of a symplectic version of a conjecture relating Gromov-Witten invariants to counts of pseudo-holomorphic curves under generic almost complex structures.

Findings

Gromov-Witten invariants are signed counts of pseudo-holomorphic curves in high degree.

Construction of Gromov-Witten pseudocycles without inhomogeneous perturbations.

Positive answer to a question by Ruan and Tian regarding pseudocycle construction.

Abstract

We prove a symplectic version of a conjecture of Lian and Pandharipande: in sufficiently high degree, the fixed-domain Gromov-Witten invariants of positive symplectic manifolds are signed counts of pseudo-holomorphic curves. The original conjecture in the complex algebraic setting was recently disproved by Beheshti et al. However, we show that the statement holds when the complex structure is replaced by a generic almost complex structure. The proof relies on showing that the fixed-domain Gromov-Witten pseudocycle can be constructed without the use of inhomogeneous or domain-dependent perturbations, which answers positively a question posed by Ruan and Tian.