Chern number identities on compact complex surfaces and applications

TL;DR

This paper establishes identities involving Chern numbers on compact complex surfaces and applies these results to characterize certain four-manifolds as Kähler surfaces under specific curvature conditions.

Contribution

It introduces new Chern number identities on compact complex surfaces and uses them to identify conditions under which a four-manifold admits a Kähler structure.

Findings

Chern number identities for compact complex surfaces

Characterization of Kähler surfaces via curvature conditions

Conditions for a four-manifold to be Kähler

Abstract

In this paper, we establish Chern number identities on compact complex surfaces. As an application, we prove that if $(M,g)$ is a compact Riemannian four-manifold with constant scalar curvature and admits a compatible complex structure $J$ such that the complexified Ricci curvature is a non-positive $(1,1)$ form, then $M$ is a K\"ahler surface.