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The volume of singular Kähler–Einstein Fano varieties

Part of: Algebraic groups Complex manifolds Surfaces and higher-dimensional varieties Local theory

Published online by Cambridge University Press:  29 April 2018

Yuchen Liu
Yuchen Liu*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ, 08544-1000, USA email yuchenl@math.princeton.edu
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Abstract

We show that the anti-canonical volume of an $n$-dimensional Kähler–Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

Keywords

Fano varietiessingularitiesK-stabilityKähler–Einstein metrics

MSC classification

Primary: 14J45: Fano varieties 14B05: Singularities
Secondary: 14L24: Geometric invariant theory 32Q15: Kähler manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 6 , June 2018 , pp. 1131 - 1158
Copyright
© The Author 2018 

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