The volume of singular Kähler-Einstein Fano varieties

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 ${\mathrm{lct}}^n\cdot \mathrm{mult}$ 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.

29 pages; comments welcome. v5: referenced corrected, to appear in Compositio Mathematica

References in corpus (2)

• K-semistability is equivariant volume minimization
• Existence of valuations with smallest normalized volume

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