Kähler-Einstein metrics with positive curvature near an isolated log terminal singularity

Journal article, 2025

We analyze the existence of Kähler-Einstein metrics of positive curvature in the neighborhood of a germ of a log terminal singularity (X,p). This boils down to solving a Dirichlet problem for certain complex Monge-Ampère equations. We establish a Moser-Trudinger inequality (Formula presented) in subcritical regimes (Formula presented) and show the existence of smooth solutions in those cases. We show that the expected critical exponent (Formula presented) can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.