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arXiv · 1605.01034

The volume of singular K\"ahler-Einstein Fano varieties

Yuchen Liu
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Abstract

We show that the anti-canonical volume of an $n$-dimensional K\"ahler-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 K\"ahler-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-Musta\c{t}\u{a} type inequality holds on its affine cone.

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Yuchen Liu. 2016-05-03. The volume of singular K\"ahler-Einstein Fano varieties. https://doi.org/10.1112/s0010437x18007042

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