Holomorphic 1-forms without zeros on K\"ahler threefolds

TL;DR

This paper classifies certain K"ahler threefolds that are fiber bundles over the circle, extending previous work and proving a conjecture in complex geometry with a detailed structural description.

Contribution

It generalizes the classification of fiber bundle structures on K"ahler threefolds from the projective to the non-projective case, and proves Kotschick's conjecture in dimension three.

Findings

Classification of K"ahler threefolds as fiber bundles over the circle.

Existence of finite étale covers with specific bundle structures.

Proof of Kotschick's conjecture in dimension 3.

Abstract

We classify all smooth compact connected K\"ahler threefolds that admit the structure of a $C^\infty$-fiber bundle over the circle. This generalizes the work of Hao and Schreieder in the projective case. In contrast to the projective case, there cannot always exist a smooth morphism to a positive-dimensional torus. Instead, we show that such a compact K\"ahler threefold admits a finite \'etale cover that is bimeromorphic to a $\mathbb{P}^1$-, $\mathbb{P}^2$-, or Hirzebruch surface-bundle over a locally trivial torus-fiber bundle over a smooth compact connected K\"ahler base. Our results prove Kotschick's conjecture in dimension 3.