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Kähler–Einstein metrics with positive curvature near an isolated log terminal singularity
- Part of
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- Journal:
- Compositio Mathematica / Volume 161 / Issue 4 / April 2025
- Published online by Cambridge University Press:
- 13 August 2025, pp. 714-755
- Print publication:
- April 2025
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- Article
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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.
