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

TL;DR

This paper establishes reverse H"older inequalities on the space of K"ahler metrics for Fano varieties, with applications to stability, singularity exponents, and complex geometric bounds.

Contribution

It introduces a reverse H"older inequality for K"ahler metrics on Fano varieties, extending to singular cases, and connects it to stability and singularity measures.

Findings

Reverse H"older inequality holds under bounded twisted Ricci potential.

Application to destabilizing geodesic rays and speed bounds.

Effective openness results for complex singularity exponents.

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.