Negative Holomorphic curvature and positive canonical bundle

TL;DR

This paper proves that projective manifolds with a Kähler metric of negative holomorphic sectional curvature necessarily have an ample canonical bundle, confirming a long-standing conjecture.

Contribution

It establishes a direct link between negative holomorphic sectional curvature and the ampleness of the canonical bundle in projective manifolds, confirming a conjecture.

Findings

Negative holomorphic sectional curvature implies ample canonical bundle

Confirms a conjecture by the second author

Advances understanding of curvature and algebraic properties of manifolds

Abstract

In this note we show that if a projective manifold admits a K\"ahler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.