On negativity of total $k$-jet curvature and ampleness of the canonical bundle

TL;DR

This paper proves that the canonical bundle of a projective complex manifold with strong negative total $k$-jet curvature is ample, advancing the understanding of Kobayashi hyperbolicity and algebraic geometry.

Contribution

It establishes the ampleness of the canonical bundle under a stronger negative jet curvature condition, linking differential geometry with algebraic properties.

Findings

Proves ampleness of $K_X$ under negative total $k$-jet curvature.

Uses positivity of direct image sheaves and jet decomposition techniques.

Produces pluridifferentials on $X$ to support the main result.

Abstract

A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle $K_X$ of a Kobayashi hyperbolic compact complex manifold $X$ is ample. In this note we prove that $K_X$ is ample if $X$ is projective and satisfies a stronger condition of nondegenerate negative total $k$-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on $X$.