# Zeros of holomorphic one-forms and topology of K\"ahler manifolds

## TL;DR

This paper explores the relationship between holomorphic one-forms without zeros and the topology of compact Kähler manifolds, confirming a conjecture in dimension two and for certain threefolds.

## Contribution

It develops a new approach to Kotschick's conjecture and verifies it for complex surfaces and some three-dimensional cases.

## Key findings

- Confirmed Kotschick's conjecture for complex surfaces.
- Proved the conjecture for smooth projective threefolds in collaboration.
- Established a link between holomorphic one-forms without zeros and fiberings over the circle.

## Abstract

A conjecture of Kotschick predicts that a compact K\"ahler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao, we use our approach to prove Kotschick's conjecture for smooth projective threefolds.