Kähler–Einstein Metrics on Stable Varieties and log Canonical Pairs

By Robert J. Berman, Henri Guenancia. Geometric and Functional Analysis, 2013

93 citations recorded by the source corpus.

Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class KX defines an ample Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document} -line bundle, and satisfying the conditions G1 and S2. Our main result says that X admits a Kähler–Einstein metric iff X has semi-log canonical singularities i.e. iff X is a stable variety in the sense of Kollár–Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kähler–Einstein metric in this singular context simply means a Kähler–Einstein on the regular locus of X with volume equal to the algebraic volume of KX, i.e. the top intersection number of KX. We also show that such a metric is uniquely determined and extends to define a canonical positive current in c1(KX). Combined with recent results of Odaka our main result shows that X admits a Kähler–Einstein metric iff X is K-stable, which thus confirms the Yau–Tian–Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kähler–Einstein metrics on quasi-projective varieties.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d19ee95452bbf21c3771f758c895a76afa0b9260

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Smooth and singular Kahler-Einstein metrics (2014, background): “Part (iii) in the case Ricωφ = Ricω + √ −1∂∂̄ψ2 − √ −1∂∂̄ψ1 is due to Pǎun [196] (whose more quantitative formulation appears in [24, 22]) together with Campana– Guenancia–Pǎun [52] that allows to assume only a lower bound on the bisectional curvature of the reference metric (without an upper bound ”
K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics (2012, background): “The opposite case of varieties polarized by KX was very recently established in [6], building on [51].”
A variational approach to the Yau–Tian–Donaldson conjecture (2015, result): “X;) klt, !further has cone singularities along the normal crossing part of [GP16]. When <0 (resp. = 0 and klt), the variational approach of [BBGZ13] provides a unique solution to (TKE) (compare [BG14]), generalizing classical results of Aubin and Yau [Aub78, Yau78]. We refer to [Tsu08, ST12, Tos1”