Alpha invariant and K-stability of \(\mathbb Q\)-Fano varieties
type: Item
instance of: scholarly article
title: Alpha invariant and K-stability of \(\mathbb Q\)-Fano varieties
zbMATH Open document ID: 1243.14037
author: Yuji Odaka
author: Yuji Sano
published in: Advances in Mathematics
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.6131
reviewed by: Julien Keller
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14J17
Mathematics Subject Classification ID: 32Q20
zbMATH DE Number: 6023755
zbMATH Keywords: alpha-invariant
zbMATH Keywords: log-canonical thresholds
zbMATH Keywords: K-stability
zbMATH Keywords: Seshadri constants
MaRDI profile type: Publication
OpenAlex ID: W2030596113
arXiv ID: 1011.6131
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cites work: The α -invariant on toric Fano manifolds
cites work: K-stability of constant scalar curvature Kähler manifolds
cites work: On Kähler-Einstein metrics on certain Kähler manifolds with \(C_ 1(M)>0\)
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cites work: Q3137105
DOI: 10.1016/J.AIM.2012.01.017
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review text: G. Tian introduced in the 80's a holomorphic invariant, the so-called \(\alpha\) invariant, to obtain a sufficient condition for existence of a Kähler-Einstein metric on a given Fano smooth complex manifold \(X\). This condition can simply be written as \(\alpha(X)>n/(n+1)\) where \(n\) is the complex dimension of \(X\). On another hand, it is expected (by the famous Yau-Tian-Donaldson conjecture) that the K-stability of the couple \((X,-K_X)\) should be equivalent to the existence of a Kähler-Einstein metric in the class \(c_1(X)>0\). Therefore it is very natural to expect that one can obtain from Tian's condition the K-stability of \((X,-K_X)\) by a direct algebro-geometric method. In a nutshell, this is what is done by the authors. It is important to remark that their method generalizes to the case of varieties over a closed field of characteristic 0.NEWLINENEWLINEIn this note, the authors prove that if the global log canonical threshold of \(X\) satisfies NEWLINE\[NEWLINElct(X)=\inf_{m\in \mathbb{N}^*} \inf_{D\sim (-mK_X)} \mathrm{lct}(X,\frac{1}{m}D) >\frac{n}{n+1},NEWLINE\]NEWLINE where \(X\) is a log-canonical \(\mathbb{Q}\)-Fano variety, then \((X,-K_X)\) is K-stable. Note that above, \(D\) varies in the set of effective divisors linearly equivalent to \(-mK_X\) and for smooth manifolds, \(\mathrm{lct}(X)=\alpha(X)\) by a result of J-P. Demailly. NEWLINENEWLINENEWLINE NEWLINEThe proof is based on the one hand on a formula for the Donaldson-Futaki invariant derived from the work of the first author and on the other hand from the relationship between the log-canonical thresholds and the Seshadri constants. A \(G\)-invariant version of the main result is given, where \(G\) is a subgroup of the automorphism group \(\mathrm{Aut}(X)\).NEWLINENEWLINEThe paper is well written.
description: scientific article; zbMATH DE number 6023755