# Frames of 1-forms on varieties and maps to abelian varieties

A fruitful question in complex algebraic geometry is how much the global 1-forms on a variety constrains its geometry. The foundational work of Popa and Schnell shows that if a variety admits a nowhere vanishing 1-form then it cannot be general type. We build off this theorem to consider varieties X admitting a frame of g everywhere independent 1-forms. This property heavily constrains the birational type of X. Under additional hypotheses that ensure X is "as general type as possible," we prove that X is a smooth isotrivial fibration over an abelian variety. Our methods also verify certain conjectures about the existence and structure of smooth maps to abelian varieties for source varieties with large Kodaira dimensions. This is joint work with Nathan Chen and Feng Hao.