Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics

TL;DR

This paper explores the relationships between Chern-Ricci curvatures, holomorphic sectional curvature, and Hermitian metrics, proving that certain curvature conditions imply Kählerness and providing examples of non-Kähler metrics with specific curvature properties.

Contribution

It introduces new formulae for curvatures of Hermitian metrics and establishes conditions under which a locally conformal Kähler manifold is Kähler, along with constructing explicit non-Kähler examples.

Findings

A compact locally conformal Kähler manifold with constant nonpositive holomorphic sectional curvature is Kähler.

Existence of complete non-Kähler metrics with pointwise negative constant holomorphic sectional curvature.

Existence of complete non-Kähler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensor.

Abstract

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is K\"{a}hler. We also give examples of complete non-K\"{a}hler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-K\"{a}hler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.