Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties
type: Item
instance of: scholarly article
title: Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties
zbMATH Open document ID: 1430.14083
author: Robert J. Berman
author: Sebastien Boucksom
author: Philippe Eyssidieux
author: Vincent Guedj
author: Ahmed Zeriahi
published in: Journal für die Reine und Angewandte Mathematik
publication date: 2019-06-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.7158
review text: Let \(X\) be a Fano manifold, that is a complex projective manifold such that the anticanonical divisor \(-K_X\) is ample. By the work of \textit{X. Chen} et al. [J. Am. Math. Soc. 28, No. 1, 199--234 (2015; Zbl 1312.53097)] we know that \(X\) admits a Kähler-Einstein metric if and only if \(X\) is \(K\)-stable. In order to define compactifications of moduli spaces of Kähler-Einstein Fano manifolds, it is necessary to generalize the theory to varieties with some mild singularities. In this paper the authors consider \(\mathbb Q\)-Fano varieties, that is normal projective varieties such that some multiple of the anticanonical divisor \(-K_X\) is Cartier and ample. They show that if \(X\) admits a Kähler-Einstein metric, the singularities of \(X\) are at most log-terminal in the sense of the minimal model program. Then they define a Mabuchi functional Mab and \(J\)-functional \(J\) for log-terminal \(\mathbb Q\)-Fano varieties, and as in the smooth setting, one says that the Mabuchi functional is proper if \(\mbox{Mab} \to \infty\) if \(J \to \infty\). The first main result of the paper generalizes classical results of \textit{S. Bando} and \textit{T. Mabuchi} [Adv. Stud. Pure Math. 10, 11--40 (1987; Zbl 0641.53065)] and \textit{W. Ding} and \textit{G. Tian} [Invent. Math. 110, No. 2, 315--335 (1992; Zbl 0779.53044)] to this singular setting: the identity component \(\mbox{Aut}^0(X)\) of the automorphism group acts transitively on the set of Kähler-Einstein metrics. Moreover, if the Mabuchi functional is proper, then \(\mbox{Aut}^0(X)=1\) and the unique minimizer of the Mabuchi functional is a Kähler-Einstein metric \(\omega_{KE}\). \newline In the second part of the paper the authors describe the behaviour of the Ricci iteration (introduced in [\textit{J. Keller}, J. Inst. Math. Jussieu 8, No. 4, 743--768 (2009; Zbl 1178.32020); \textit{Y. A. Rubinstein}, Adv. Math. 218, No. 5, 1526--1565 (2008; Zbl 1143.53065)]) and the Kähler-Ricci flow (see [\textit{J. Song} and \…
reviewed by: Andreas Höring
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 32Q20
Mathematics Subject Classification ID: 53E30
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 7062932
zbMATH Keywords: Kähler-Einstein metric
zbMATH Keywords: Fano variety
zbMATH Keywords: Kähler-Ricci flow
zbMATH Keywords: Ricci iteration
zbMATH Keywords: Mabuchi functional
Wikidata QID: Q103487178
MaRDI profile type: Publication
OpenAlex ID: W2963754085
arXiv ID: 1111.7158
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Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties (DOI)
DOI: 10.1515/CRELLE-2016-0033