# Direct images of pseudoeffective cotangent bundles

## TL;DR

This paper explores the positivity properties of cotangent bundles on compact Kähler manifolds, generalizing known results from elliptic curves and applying them to abelian fibrations with pseudoeffective cotangent bundles.

## Contribution

It generalizes the Demailly-Peternell-Schneider result to arbitrary compact Kähler manifolds and discusses implications for abelian fibrations with pseudoeffective cotangent bundles.

## Key findings

- The tautological class of the Serre bundle on an elliptic curve contains a unique positive current.
- Generalization of positivity results to broader classes of Kähler manifolds.
- Open questions on nonvanishing properties of pseudoeffective cotangent bundles.

## Abstract

Let A be an elliptic curve, and let $V_A$ be the Serre vector bundle on A. A famous example of Demailly-Peternell-Schneider shows that the tautological class of $V_A$ contains a unique closed positive current. In this survey we start by generalising this statement to arbitrary compact K\"ahler manifolds. We then give an application to abelian fibrations $X \rightarrow Y$ where the total space X has pseudoeffective cotangent bundle and raise some questions about nonvanishing properties of these bundles.