An extension of a theorem of Wu-Yau

TL;DR

This paper proves that compact Kähler manifolds with nonpositive holomorphic sectional curvature have nef canonical bundles, and those with negative curvature have ample canonical bundles, confirming a conjecture of Yau.

Contribution

It extends Wu-Yau's recent results to a broader class of Kähler manifolds, establishing the nefness and ampleness of the canonical bundle based on curvature conditions.

Findings

Manifolds with nonpositive curvature have nef canonical bundles.

Negative curvature implies the canonical bundle is ample.

Confirms Yau's conjecture in the Kähler setting.

Abstract

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu-Yau.