A Poisson-Kähler fibration has a canonical Kähler metric on its base whose holomorphic bisectional curvature is non-positive and whose holomorphic sectional, Ricci, and scalar curvatures are bounded above by a negative constant.
A curvature formula associated to a family of pseudoconvex domains
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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.
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Curvature of the base manifold of a Monge-Amp\`ere fibration and its existence
A Poisson-Kähler fibration has a canonical Kähler metric on its base whose holomorphic bisectional curvature is non-positive and whose holomorphic sectional, Ricci, and scalar curvatures are bounded above by a negative constant.