2023/09/28 by Robert J. Berman, Berman, Robert J.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2309.16278
openalex publication_date 2023/09/28 · openalex created_date 2023/10/01 · openalex updated_date 2026/07/28
A reverse Hölder inequality is established on the space of Kähler metrics in the first Chern class of a Fano manifold X endowed with Darvas Lp-Finsler metrics. The inequality holds under a uniform bound on a twisted Ricci potential and extends to Fano varieties with log terminal singularities. Its proof leverages a "hidden" log-concavity. An application to destabilizing geodesic rays is provided, which yields a reverse Hölder inequality for the speed of the geodesic. In the case of Aubin's continuity path on a K-unstable Fano variety, the constant in the corresponding Hölder bound is shown to only depend on p and the dimension of X. This leads to some intruiging relations to Harnack bounds and the partial C0-estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of ω-plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions.