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 solve a Dirichlet problem for certain complex Monge-Ampère equations. We show that the solvability of the latter is independent of the shape of the domain and of the boundary data. We establish a Moser-Trudinger $(MT{)}_{\gamma }$ inequality in subcritical regimes $\gamma <{\gamma }_p$ and establish the existence of smooth solutions in that cases. We show that the expected critical exponent ${\hat{\gamma }}_p=\frac{n+1}{n}\widehat{\mathrm{vol}}(X,p{)}^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.

39 pages, no figure. Appendix by Sébastien Boucksom

References in corpus (1)

• Finite generation for valuations computing stability thresholds and applications to K-stability