Reverse Hölder inequalities on the space of Kähler metrics of a Fano variety and effective openness

Journal article, 2025

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’ 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 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 intriguing 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.