Nowhere vanishing holomorphic one-forms on varieties of Kodaira codimension one

TL;DR

This paper investigates the conditions under which complex varieties of Kodaira dimension one admit holomorphic 1-forms without zeros, revealing a link to morphisms onto elliptic curves and providing a structural classification.

Contribution

It establishes a characterization of varieties of Kodaira dimension one that admit zero-free holomorphic 1-forms, connecting this property to morphisms onto elliptic curves and offering a structure theorem.

Findings

A minimal smooth projective variety of Kodaira dimension n-1 admits a zero-free holomorphic 1-form iff it maps onto an elliptic curve.

Provides a structure theorem for non-minimal varieties of Kodaira codimension one with zero-free holomorphic 1-forms.

Extends the understanding of zeros of holomorphic 1-forms to varieties of intermediate Kodaira dimension.

Abstract

Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal smooth projective variety $X$ of Kodaira dimension $\kappa(X)=\dim X-1$ admits a holomorphic 1-form without zero if and only if there is a smooth morphism from $X$ to an elliptic curve. Furthermore, for a general smooth projective variety (not necessarily minimal) $X$ of Kodaira codimension one, we give a structure theorem for $X$ given that $X$ admits a holomorphic 1-form without zero.