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

Simon Pietig

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arXiv · 2506.22067

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

Simon Pietig

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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.

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Simon Pietig. 2025-06-27. Holomorphic 1-forms without zeros on K\"ahler threefolds. https://arxiv.org/abs/2506.22067

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