Generic transversality for unbranched covers of closed pseudoholomorphic curves

TL;DR

This paper demonstrates that generic perturbations of the almost complex structure in closed manifolds ensure transversality for unbranched covers of pseudoholomorphic curves, enabling integer-valued Gromov-Witten invariants in certain cases.

Contribution

It establishes a transversality result for unbranched covers of pseudoholomorphic curves using analytic perturbation techniques, facilitating computation of Gromov-Witten invariants as signed counts.

Findings

Transversality achieved for all unbranched multiple covers with index zero.

Gromov-Witten invariants can be computed as integer counts for generic tame almost complex structures.

The method relies on an analytic perturbation technique originally due to Taubes.

Abstract

We prove that in closed almost complex manifolds of any dimension, generic perturbations of the almost complex structure suffice to achieve transversality for all unbranched multiple covers of simple pseudoholomorphic curves with deformation index zero. A corollary is that the Gromov-Witten invariants (without descendants) of symplectic 4-manifolds can always be computed as a signed and weighted count of honest J-holomorphic curves for generic tame J: in particular, each such invariant is an integer divided by a weighting factor that depends only on the divisibility of the corresponding homology class. The transversality proof is based on an analytic perturbation technique, originally due to Taubes.