Positivity of cotangent bundles

TL;DR

This paper generalizes a criterion for the positivity of cotangent bundles from surfaces to higher-dimensional projective varieties, introducing a new notion called quasi-ampleness.

Contribution

It extends Schneider's theorem by defining quasi-ampleness and applying it to higher-dimensional cases, broadening the understanding of cotangent bundle positivity.

Findings

Established a criterion for quasi-ampleness of cotangent bundles in higher dimensions.

Extended Schneider's theorem to complex projective varieties beyond surfaces.

Introduced and utilized the concept of quasi-ampleness as a weaker form of ampleness.

Abstract

In this paper we prove a generalization of a theorem of Schneider, which gives a criterion for a projective surface over the complex numbers to have an ample cotangent bundle. After reviewing different notions of positivity, we introduce a slightly weaker notion of ampleness, which we call quasi-ample, and then are able to extend Schneider's result to higher dimensions.