A curvature formula associated to a family of pseudoconvex domains

Abstract: We shall give a definition of the curvature operator for a family of weighted Bergman spaces ${\mathcal H_t}$ 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 ${\partial D_t}$. As an application, we get a variation formula for the norms of Bergman projections of currents with compact support. A flatness criterion for ${\mathcal H_t}$ and its applications to triviality of fibrations are also given in this paper.