Generic Transversality for Unbranched Covers of Closed Pseudoholomorphic Curves

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 0. 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.© 2016 Wiley Periodicals, Inc.

Seiberg–Witten and Gromov invariants for self-dual harmonic 2–forms

“…Acknowledgements. The main idea stems from a chat with Cliff Taubes on how to extend his transversality results from closed symplectic manifolds to symplectic cobordisms, and builds off of related work with Chris Wendl on closed symplectic manifolds [10]. I appreciate them and their inspiration.…”

Conjugate linear perturbations of Dirac operators and Majorana fermions

“…Furthermore, in certain cases the above operators have been studied in the context of pseudo-holomorphic curves; cf. [6,7,9,12,18,20,21].…”