Tamed to compatible: Symplectic forms via moduli space integration

TL;DR

This paper demonstrates that on certain 4-manifolds, the space of tamed almost complex structures contains a dense subset of compatible structures, constructed via integration over pseudo-holomorphic curves.

Contribution

It proves that tamed almost complex structures can be approximated by compatible ones using a novel integration method over currents defined by pseudo-holomorphic curves.

Findings

Dense subset of compatible structures in the space of tamed structures

Construction of symplectic forms via integration over pseudo-holomorphic curves

Compatibility achieved within the same cohomology class

Abstract

Fix a compact 4-dimensional manifold with self-dual 2nd Betti number one and with a given symplectic form. This article proves the following: The Frechet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.