Abstract 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.
Benjamin Church (Wed,) studied this question.