Perturbations of the metric in Seiberg-Witten equations

Abstract

Let $M$ a compact connected orientable 4-manifold. We study the space $\Xi$ of $Spin^c$-structures of fixed fundamental class, as an infinite dimensional principal bundle on the manifold of riemannian metrics on $M$. In order to study perturbations of the metric in Seiberg-Witten equations, we study the transversality of universal equations, parametrized with all $Spin^c$-structures $\Xi$. We prove that, on a complex K\"ahler surface, for an hermitian metric $h$ sufficiently close to the original K\"ahler metric, the moduli space of Seiberg-Witten equations relative to the metric $h$ is smooth of the expected dimension.