Zeros of one-forms and homologically trivial fibrations

TL;DR

This paper proves a conjecture about one-forms without zeros on certain compact Kähler manifolds by linking it to a conjecture on homologically trivial fibrations, providing new insights into the structure of these manifolds.

Contribution

It establishes the validity of Kotschick's conjecture for a class of compact Kähler manifolds with simple Albanese torus and high first Betti number, connecting it to existing conjectures.

Findings

Kotschick's conjecture holds for manifolds with simple Albanese torus and high first Betti number.

The conjecture of Bobadilla and Kollár implies Kotschick's conjecture in this setting.

The paper extends understanding of the structure of Kähler manifolds with specific topological properties.

Abstract

We show that a conjecture of Kotschick about one-forms without zeros on compact K\"ahler manifolds follows in the case of simple Albanese torus from a conjecture of Bobadilla and Koll\'ar about homologically trivial fibrations. As an application, we prove Kotschick's conjecture for compact K\"ahler manifolds X whose first betti number is at least 2dim(X)-2 and Albanese torus is simple.