Direct images of pseudoeffective cotangent bundles

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 Kähler manifolds. We then give an application to abelian fibrations $X\to Y$ where the total space X has pseudoeffective cotangent bundle and raise some questions about nonvanishing properties of these bundles.