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

Vincent Guedj; Antonio Trusiani; Sébastien Boucksom

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arXiv · 2306.07900

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

Vincent Guedj·Antonio Trusiani·Sébastien Boucksom

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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 $(MT)_γ$ inequality in subcritical regimes $γ<γ_p$ and establish the existence of smooth solutions in that cases. We show that the expected critical exponent $\hatγ_p=\frac{n+1}{n} \widehat{\mathrm{vol}}(X,p)^{1/n}$ can be expressed in terms of the normalized volume, an important algebraic invariant of the singularity.

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BibTeX RIS

Vincent Guedj, Antonio Trusiani, Sébastien Boucksom. 2023-06-13. Kähler-Einstein metrics with positive curvature near an isolated log terminal singularity. https://arxiv.org/abs/2306.07900

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