  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.
