Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle

TL;DR

This paper proves that compact complex manifolds with a Kähler metric of non-positive holomorphic sectional curvature, which is strictly negative somewhere, necessarily have a positive canonical bundle, linking curvature to algebraic positivity.

Contribution

It establishes a new connection between curvature conditions and the positivity of the canonical bundle in complex geometry.

Findings

Manifolds with such curvature conditions have positive canonical bundles.

The result applies to compact complex manifolds with specific curvature properties.

It advances understanding of the relationship between curvature and algebraic geometry.

Abstract

We show that if a compact complex manifold admits a K\"ahler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.