On the fixed domain Gromov-Witten invariants of positive symplectic manifolds

Abstract: Using pseudo-holomorphic curves, we establish a new enumerativity result for the fixed domain Gromov-Witten invariants and prove a symplectic version of a conjecture of Lian and Pandharipande. The original conjecture, which asserts that these invariants are enumerative for projective Fano manifolds and high degree curves, was recently disproved by Beheshti et al. However, we show that it holds when a complex structure is replaced by a generic almost complex structure. Our result explains the integrality of the fixed domain Gromov-Witten invariants observed in examples by Buch and Pandharipande.