August 17, 2025

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

Key Points

Kähler–Einstein metrics of positive curvature exist near isolated log terminal singularities, indicating new geometric insights.
The study establishes a Moser–Trudinger inequality in subcritical regimes, facilitating the solution of complex equations.
Smooth solutions are shown to exist for the Dirichlet problem in the presence of specific conditions, enhancing previous theory.
Understanding the expected critical exponent connects geometry to algebraic properties, highlighting the significance of normalized volume.

Abstract

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.

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