On the Scalar Curvature of Complex Surfaces

TL;DR

This paper establishes a link between the existence of positive scalar curvature metrics and Kähler metrics on minimal compact complex surfaces, extending prior results by Witten and Kronheimer.

Contribution

It proves that a minimal compact complex surface admits a positive scalar curvature metric if and only if it admits a Kähler metric with positive scalar curvature.

Findings

Equivalence between positive scalar curvature and Kähler positive scalar curvature on complex surfaces.

Extension of Witten and Kronheimer's results to minimal compact complex surfaces.

Abstract

Let (M,J) be a minimal compact complex surface of Kaehler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a KAEHLER metric of positive scalar curvature. This extends previous results of Witten and Kronheimer.