415,785 research outputs found

    A remark on the Alexandrov-Fenchel inequality

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    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 L2L^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

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    We shall give a definition of the curvature operator for a family of weighted Bergman spaces {Ht}\{\mathcal H_t\} associated to a smooth family of smoothly bounded strongly pseudoconvex domains {Dt}\{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 {∂Dt}\{\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 {Ht}\{\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

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