Nowhere vanishing holomorphic one-forms and fibrations over abelian varieties

TL;DR

This paper classifies complex projective varieties with maximal holomorphic 1-forms, showing they admit fibrations over abelian varieties and are constructed as quotients of products, extending previous results and conjectures.

Contribution

It proves that varieties with maximal holomorphic 1-forms admit smooth morphisms to abelian varieties and classifies their structure as diagonal quotients, extending known cases.

Findings

Varieties with maximal holomorphic 1-forms admit fibrations over abelian varieties.

Such varieties are classified as diagonal quotients of products involving abelian varieties.

A birational classification is established without minimality assumptions.

Abstract

A result of Popa and Schnell shows that any holomorphic 1-form on a smooth complex projective variety of general type admits zeros. More generally, given a variety $X$ which admits $g$ pointwise linearly independent holomorphic 1-forms, their result shows that $X$ has Kodaira dimension $\kappa(X) \leq \dim X - g$. In the extremal case where $\kappa(X) = \dim X - g$ and $X$ is minimal, we prove that $X$ admits a smooth morphism to an abelian variety, and classify all such $X$ by showing they arise as diagonal quotients of the product of an abelian variety with a variety of general type. The case $g = 1$ was first proved by the third author, and classification results about surfaces and threefolds carrying nowhere vanishing forms have appeared in work of Schreieder and subsequent joint work with the third author. We also prove a birational version of this classification which holds without the minimal assumption, and establish additional cases of a conjecture of the third author.