  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.
