Perturbations of the metric in Seiberg-Witten equations

TL;DR

This paper investigates how perturbing the metric affects the Seiberg-Witten equations on 4-manifolds, establishing conditions under which the moduli space remains smooth and of expected dimension.

Contribution

It introduces a framework for studying metric perturbations in Seiberg-Witten equations and proves smoothness of the moduli space near Kähler metrics.

Findings

Moduli space is smooth for metrics close to the original Kähler metric.

Transversality of universal equations is established for all $Spin^c$-structures.

Results apply to complex Kähler surfaces with hermitian metrics.

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.