Holomorphic 1-forms without zeros on Kähler threefolds

2025/06/27 by Simon Pietig, Pietig, Simon · 1 citation
Mathematics · #32J17 #32Q15 #32Q55 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2506.22067

openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify all smooth compact connected Kähler threefolds that admit the structure of a C^∞-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 Kähler threefold admits a finite étale cover that is bimeromorphic to a ℙ¹-, ℙ²-, or Hirzebruch surface-bundle over a locally trivial torus-fiber bundle over a smooth compact connected Kähler base. Our results prove Kotschick's conjecture in dimension 3.

Citations

• Nowhere vanishing holomorphic one-forms on varieties of Kodaira codimension one
• Zeros of holomorphic one-forms and topology of Kähler manifolds
• Smooth Families of Tori and Linear Kähler Groups
• Mori fibre spaces for Kähler threefolds
• Minimal models for Kähler threefolds
• Kodaira dimension and zeros of holomorphic one-forms
• Q.E.D. for algebraic varieties
• Holomorphic 2-forms on Complex Threefolds

Cited by

• Holomorphic one-forms without zeros on Kähler manifolds of Kodaira codimension one

Related

• Zeros of holomorphic one-forms and topology of Kähler manifolds
• Compact Kähler 3-manifolds without non-trivial subvarieties
• Zeros of one-forms and the topology of algebraic maps
• Algebraic approximation of Kähler threefolds
• Compact K" ahler threefolds of π₁-general type