A remark on the Alexandrov-Fenchel inequality

In this article, we give a complex-geometric proof of the Alexandrov-Fenchel inequality without using toric compactifications. The idea is to use the Legendre transform and develop the Brascamp-Lieb proof of the Pr\'ekopa theorem. New ingredients in our proof include an integration of Timorin's mixed Hodge-Riemann bilinear relation and a mixed norm version of H\"ormander's $L^2$-estimate, which also implies a non-compact version of the Khovanski\u{i}-Teissier inequality.Comment: New version, "on line first" in Journal of Functional Analysis: https://doi.org/10.1016/j.jfa.2018.01.01

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.Comment: 35 pages, to appear in Annales de l'Institut Fourie

Bergman completeness is not a quasi-conformal invariant

We show that Bergman completeness is not a quasi-conformal invariant for general Riemann surfaces