Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs
Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 743-841.

The purpose of this paper is to set up a formalism inspired by non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat–Heckman measures in the context of test configurations for arbitrary polarized schemes, characterizing in particular almost trivial test configurations. Second, for any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study non-Archimedean analogues of certain classical functionals in Kähler geometry. These functionals are defined on the space of test configurations, and the Donaldson–Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and singularities of pairs, reproving and strengthening Y. Odaka’s results in our formalism. This provides various examples of uniformly K-stable varieties.

Le but de cet article est de mettre en place un formalisme inspiré par la géométrie non-archimédienne pour étudier la K-stabilité. Nous fournissons d’abord une analyse détaillée des mesures de Duistermaat–Heckman dans le contexte des configurations test de schémas polarisés arbitraires, caractérisant en particulier les configurations test presque triviales. Nous introduisons et étudions ensuite, pour toute variété normale polarisée (ou, plus généralement, toute paire polarisée au sens du Programme du Modèle Minimal), les analogues non-archimédiens de certaines fonctionnelles classiques de la géométrie kählérienne. Ces fonctionnelles sont définies sur l’espace des configurations test, et l’invariant de Donaldson–Futaki est en particulier interprété comme la version non-archimédienne de la fonctionnelle de Mabuchi, à un terme d’erreur explicite près. Enfin, nous étudions en détail les liens entre K-stabilité uniforme et singularités des paires, redémontrant et améliorant des résultats de Y. Odaka dans notre formalisme. Ceci fournit divers exemples de variétés uniformément K-stables.

Published online:
DOI: 10.5802/aif.3096
Classification: 14L30, 14E30
Keywords: K-stability, Duistermaat–Heckman measures, singularities of pairs.
Boucksom, Sébastien 1; Hisamoto, Tomoyuki 2; Jonsson, Mattias 3

1 CNRS-CMLS École Polytechnique 91128 Palaiseau Cedex (France)
2 Graduate School of Mathematics Nagoya University Furocho Chikusa Nagoya (Japan)
3 Dept of Mathematics University of Michigan Ann Arbor, MI 48109–1043 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Boucksom, Sébastien; Hisamoto, Tomoyuki; Jonsson, Mattias. Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs. Annales de l'Institut Fourier, Volume 67 (2017) no. 2, pp. 743-841. doi : 10.5802/aif.3096. https://aif.centre-mersenne.org/articles/10.5802/aif.3096/

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