Nowhere vanishing 1-forms on four-folds

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