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1 November 2017 K-semistability is equivariant volume minimization
Chi Li
Duke Math. J. 166(16): 3147-3218 (1 November 2017). DOI: 10.1215/00127094-2017-0026

Abstract

This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over Q-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler–Einstein metrics on Fano varieties. In particular, we prove that for a Q-Fano variety V, the K-semistability of (V,KV) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ordV among all C-invariant valuations on the cone associated to any positive Cartier multiple of KV. In this case, we show that ordV is the unique minimizer among all C-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V.

Citation

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Chi Li. "K-semistability is equivariant volume minimization." Duke Math. J. 166 (16) 3147 - 3218, 1 November 2017. https://doi.org/10.1215/00127094-2017-0026

Information

Received: 5 May 2016; Revised: 31 March 2017; Published: 1 November 2017
First available in Project Euclid: 16 September 2017

zbMATH: 06812216
MathSciNet: MR3715806
Digital Object Identifier: 10.1215/00127094-2017-0026

Subjects:
Primary: 14B05
Secondary: 13A18 , 52A27 , 53C25

Keywords: Ding semistability , Kähler–Einstein metrics , K-semistability , normalized volume , real valuations , volume minimization

Rights: Copyright © 2017 Duke University Press

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Vol.166 • No. 16 • 1 November 2017
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