# Holomorphic one-forms without zeros on threefolds

## TL;DR

This paper characterizes smooth complex projective threefolds that admit a holomorphic one-form without zeros, linking this property to the topology of the underlying real 6-manifold and providing a complete classification, thus confirming a conjecture of Kotschick.

## Contribution

It provides a full classification of threefolds with holomorphic one-forms without zeros, establishing a topological criterion and proving Kotschick's conjecture in dimension three.

## Key findings

- Threefolds admit zero-free holomorphic one-forms iff their underlying 6-manifold fibers over the circle.
- Complete classification of such threefolds.
- Proof of Kotschick's conjecture in dimension three.

## Abstract

We show that a smooth complex projective threefold admits a holomorphic one-form without zeros if and only if the underlying real 6-manifold fibres smoothly over the circle, and we give a complete classification of all threefolds with that property. Our results prove a conjecture of Kotschick in dimension three.