Geometry of Holomorphic One-forms on Smooth Projective Varieties

In this article, we show that any morphism $f$ from a smooth projective variety $X$ to a simple abelian variety $A$ is smooth, if and only if there exists a holomorphic 1-form $\omega$ on $A$ such that $f^*\omega$ has no zero. As the key ingredient in the proof, we show any $\mathbb{Z}$-homology fibre bundle morphism is without blow-up in codimension 0 in the sense of Sabbah. Furthermore, we investigate the structure of the spaces of holomorphic 1-forms with zeros, and show that they are linear for large classes of varieties. Also, we construct a delicate example of a smooth projective subvariety of an abelian variety for which the holomorphic 1-forms with positive dimensional zero loci do not form a linear subset. Finally, we study algebraic surfaces admitting holomorphic 1-forms that have zeros and do not arise from cohomology jump loci.