Compactness and rigidity of K\"{a}hler surfaces with constant scalar curvature

TL;DR

This paper establishes compactness, splitting, and rigidity results for Kähler surfaces with constant scalar curvature and for Einstein-Maxwell systems, under various geometric bounds.

Contribution

It proves a compactness theorem for Kähler surfaces with specific curvature and volume bounds, and introduces new splitting and rigidity theorems for Einstein-Maxwell systems.

Findings

Proved a compactness theorem for Kähler surfaces with bounded curvature and volume.

Established splitting theorems for Einstein-Maxwell systems.

Derived rigidity results under geometric bounds.

Abstract

A compactness theorem is proved for a family of K\"{a}hler surfaces with constant scalar curvature and volume bounded from below, diameter bounded from above, Ricci curvature bounded and the signature bounded from below. Furthermore, a splitting theorem and some rigidity theorems are proved for Einstein-Maxwell systems.