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

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)γ in subcritical regimes γ<γcrit(X,p) and show the existence of smooth solutions in those cases. We show that the expected critical exponent γ̃crit(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.