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

Abstract 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’ $$L^{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -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 $$C^{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> -estimate. In another direction, universal effective openness results are established for the complex singularity exponents (log canonical thresholds) of $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -plurisubharmonic functions on any Fano variety. Finally, another application to K-unstable Fano varieties is given, involving Archimedean Igusa zeta functions.