Quasi-projective manifolds with negative holomorphic sectional curvature

TL;DR

This paper explores the geometric properties of manifolds with negative holomorphic sectional curvature, establishing their projectivity, the general type of subvarieties, and extending results to quasi-projective cases.

Contribution

It proves that irreducible subvarieties of such manifolds are of general type and extends the results to quasi-negative curvature and quasi-projective manifolds.

Findings

Subvarieties of negatively curved manifolds are of general type.

Extension of results to quasi-negative curvature cases.

Quasi-projective manifolds with negative curvature are of log general type.

Abstract

Let $(M,\omega)$ be a compact K\"ahler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the quasi-negative curvature case building on earlier results of Diverio-Trapani. Finally, we investigate the more general setting of a quasi-projective manifold $X^{\circ}$ endowed with a K\"ahler metric with negative holomorphic sectional curvature and we prove that such a manifold $X^{\circ}$ is necessarily of log general type.