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

Abstract 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 (MT) _ in subcritical regimes < ₂ₑ₈ₓ (X, p) and show the existence of smooth solutions in those cases. We show that the expected critical exponent ₂ₑ₈ₓ (X, p) = ( (n+1) /n) vol (X, p) ^1/n can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.