A curvature formula associated to a family of pseudoconvex domains

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.