Kahler structure of the total space near a Kahler fiber

Motivated by the Kodaira-Spencer local stability theorem for Kahler structures and by C. Li’s study of Kahler structures on holomorphic submersions between compact complex manifolds, we establish an equivalent characterization of the Kahlerness of the total space near a Kahler fiber, in an optimal manner. The proof combines a (1,1)-type lifting argument for flat sections of a local system, observations on torsion-freeness and a Kahler neighborhood criterion for compact Kahler submanifolds.