Kähler structures for holomorphic submersions

Abstract

In this short paper, for any holomorphic submersion $\pi :X\to B$, we derive a criterion for $X$ to have Kähler 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 Kähler fibers and Kähler bases implies that the total spaces are Kähler. This class includes isotrivial submersions and torus fibrations.

17 pages, comments very welcome