K\"{a}hler structures for holomorphic submersions

TL;DR

This paper establishes a new criterion for when a complex manifold with a holomorphic submersion admits a Kähler structure, extending previous results and applying to specific classes like isotrivial submersions and torus fibrations.

Contribution

It generalizes Blanchard's criterion for Kähler structures in holomorphic submersions and applies it to answer a question of Harvey-Lawson for fiber dimension one.

Findings

Derived a generalized criterion for Kähler structures in holomorphic submersions.

Proved that certain Hermitian-Symplectic structures imply the total space is Kähler.

Applied results to isotrivial submersions and torus fibrations.

Abstract

In this short paper, for any holomorphic submersion $\pi: X\rightarrow B$, we derive a criterion for $X$ to have K\"{a}hler structures. This criterion generalizes Blanchard's criterion for a special class of isotrivial holomorphic submersions. We use this criterion to answer a question of Harvey-Lawson in the case of fiber dimension one. As the main application, we prove that the existence of Hermitian-Symplectic structures on certain class of holomorphic submersions with K\"{a}hler fibers and K\"{a}hler bases implies that the total spaces are K\"{a}hler. This class includes isotrivial submersions and torus fibrations.