Reverse H\"older inequalities on the space of K\"ahler metrics of a Fano variety and effective openness

Abstract

A reverse H\"older inequality is established on the space of K\"ahler metrics in the first Chern class of a Fano manifold X endowed with Darvas L^{p}-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 H\"older 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 H\"older 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 C^{0}-estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of \omega-plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions.