Hermitian manifolds with quasi-negative curvature

TL;DR

This paper demonstrates that under a specific Hermitian curvature flow, non-positive Chern-Ricci curvature remains preserved on manifolds with non-positive bisectional curvature, leading to ampleness of the canonical line bundle.

Contribution

It introduces a new Hermitian curvature flow that preserves non-positive Chern-Ricci curvature and applies this to prove ampleness of the canonical bundle.

Findings

Non-positivity of Chern-Ricci curvature is preserved under the flow.

Canonical line bundle is ample for certain Hermitian manifolds.

Flow provides new tools for studying curvature and line bundle properties.

Abstract

In this work, we show that along a particular choice of Hermitian curvature flow, the non-positivity of Chern-Ricci curvature will be preserved if the initial metric has non-positive bisectional curvature. As an application, we show that the canonical line bundle of a compact Hermitian manifold with nonpositive bisectional curvature and quasi-negative Chern-Ricci curvature is ample.