Holomorphic 1-forms without zeros on Kähler threefolds

We classify all smooth compact connected Kähler threefolds that admit the structure of a $C^{\mathrm{\infty }}$-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 Kähler threefold admits a finite étale 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 Kähler base. Our results prove Kotschick’s conjecture in dimension 3.