Geometric stability of the cotangent bundle and the universal cover of a projective manifold

TL;DR

This paper investigates the geometric properties of projective manifolds, establishing conditions under which the manifold is of general type and analyzing the Kodaira dimension related to the cotangent bundle and its quotients.

Contribution

It provides new criteria linking the cotangent bundle's subsheaves and the manifold's Kodaira dimension, extending understanding of the universal cover and stability conditions.

Findings

If a wedge power of the cotangent bundle contains a subsheaf with maximal Kodaira dimension, then the manifold is of general type.

The determinant bundle of a quotient of the cotangent bundle is always pseudo-effective.

If the canonical bundle is numerically equivalent to an effective Q-divisor, then the Kodaira dimension is non-negative.

Abstract

Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira dimension. Then we prove that X is of general type. More generally we compute the Kodaira dimension if the determinant bundle has sufficiently large Kodaira dimension. This is based on the study of the determinant bundle of a quotient of the cotangent bundle of a non-uniruled manifold: this bundle is always pseudo-effective. We apply this to study the universal cover of a projective manifold. Finally we prove the following: if the canonical bundle is numerically equivalent to an effective Q-divisor, then the Kodaira dimension is non-negative.