Remark on quasi-negative $k$-Ricci curvature

TL;DR

This paper proves that compact Kähler manifolds with quasi-negative Ricci curvature have ample canonical bundles, extending previous results by establishing positivity of the canonical bundle through rational curve analysis.

Contribution

It demonstrates that $k$-quasi-negative Ricci curvature implies the ampleness of the canonical bundle, completing a prior result by Chu-Lee-Tam.

Findings

If $X$ has $k$-quasi-negative Ricci curvature, then $K_X$ is ample.

Any rational curve on $X$ satisfies $K_X ullet C > 0$.

The result extends the understanding of curvature conditions leading to positivity of the canonical bundle.

Abstract

In this note, we show that if $X$ is a compact K\"ahler manifold with $k$-quasi-negative Ricci curvature, then $K_X$ is ample. This completes a previous result of Chu-Lee-Tam. The key ingredient is an observation that any rational curve $C$ on $X$ satisfy $K_X\cdot C>0$ based on the Gauss-Codazzi equation.