Nowhere vanishing 1-forms on 4-folds

Abstract

In this note, we prove -- in dimension at most 4 -- a conjectue of Hao which says that a morphism $f : X \to A$ to a simple abelian variety $A$ is smooth if and only if there is a 1-form pulled back from A without any zeros. We also give a complete classification of 4-folds with a 1-form without zeros not admitting a map to an elliptic curve.