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

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.

v2: main theorem improved to incorporate arbitrary constraints; added discussion of the Ruan-Tian and McDuff-Salamon construction of the Gromov-Witten pseudocycle; changed title, abstract, and introduction; 36 pages