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 $(MT)_{\gamma}$ in subcritical regimes $\gamma<\gamma_{\rm crit}(X,p)$ and show the existence of smooth solutions in those cases. We show that the expected critical exponent $\tilde{\gamma}_{\rm crit}(X,p)=(({n+1})/{n}) \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.