Moser-Trudinger type inequalities for complex Monge-Amp\`ere operators and Aubin's "hypoth\`ese fondamentale"

TL;DR

This paper establishes Moser-Trudinger type inequalities for complex Monge-Ampère operators on Kähler manifolds, confirming Aubin's hypothesis and exploring sharpness, applications, and counterexamples in complex geometry.

Contribution

It proves Aubin's fundamental hypothesis for these inequalities on Kähler manifolds and analyzes their sharpness and applications in complex Monge-Ampère equations.

Findings

Inequalities hold on any integral compact Kähler manifold.

Constants depend only on the dimension in the Fano case.

Sharp inequalities are established for S^1-invariant functions.

Abstract

We prove Aubin's "Hypothese fondamentale" concerning the existence of Moser-Trudinger type inequalities on any integral compact K\"ahler manifold X. In the case of the anti-canonical class on a Fano manifold the constants in the inequalities are shown to only depend on the dimension of X (but there are counterexamples to the precise value proposed by Aubin). In the different setting of pseudoconvex domains in complex space we also obtain a quasi-sharp version of the inequalities and relate it to Brezis-Merle type inequalities. The inequalities are shown to be sharp for S^{1}-invariant functions on the unit-ball. We give applications to existence and blow-up of solutions to complex Monge-Amp\`ere equations of mean field (Liouville) type.