Kähler structures for holomorphic submersions

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.