Seiberg-Witten equations on surfaces of logarithmic general type

TL;DR

This paper investigates solutions to the Seiberg-Witten equations on certain complex surfaces, providing methods to construct solutions, bounds on scalar curvature, and non-existence results for Einstein metrics on blow-ups.

Contribution

It introduces a way to construct irreducible solutions on surfaces of logarithmic general type and establishes bounds and non-existence results related to Einstein metrics.

Findings

Constructed irreducible solutions for asymptotic Poincaré type metrics.

Derived lower bounds for the $L^{2}$-norm of scalar curvature.

Proved non-existence of Einstein metrics on blow-ups of these surfaces.

Abstract

We study the Seiberg-Witten equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the Seiberg-Witten equations for any metric which is "asymptotic" to a Poincar\'e type metric at infinity. Then we compute a lower bound for the $L^{2}$-norm of scalar curvature on these spaces and give non-existence results for Einstein metrics on blow-ups.