Abstract
We shall give a definition of the curvature operator for a family of weighted Bergman spaces {H-t} associated to a smooth family of smoothly bounded strongly pseudoconvex domains {D-t}. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature for the associated family of boundaries {delta D-t} As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for {H(t)1 and its applications to triviality of fibrations are also given in this paper.
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