Variation of the Bergman kernels under deformations of complex structures

TL;DR

This paper derives a local variation formula for Bergman kernels under complex structure deformations, linking their behavior to curvature and Lie derivatives, with applications to holomorphic motions.

Contribution

It provides a new local variation formula for Bergman kernels in deformed complex manifolds, extending previous work with a Lie derivative approach.

Findings

Derived a local variation formula for Bergman kernels

Connected Bergman kernel behavior to curvature and Lie derivatives

Provided a criterion for triviality of holomorphic motions

Abstract

Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct image sheaf. In general, following Schumacher's idea, we use the Lie derivative to compute the variation. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.