A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations

Boucksom, Sébastien; Jonsson, Mattias

doi:10.5802/aif.3668

A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations
[Une approche non-archimédienne à la K-stabilité, I : géométrie métrique des espaces de configurations test et de valuations]

Boucksom, Sébastien ¹ ; Jonsson, Mattias ²

¹ Sorbonne Université and Université Paris Cité CNRS, IMJ-PRG F-75005 Paris (France)
² Dept of Mathematics University of Michigan Ann Arbor, MI 48109-1043 (USA)

Annales de l'Institut Fourier, Online first, 99 p.

Résumé
Abstract

Pour toute variété polarisée $(X, L)$ , nous montrons que les configurations test, et plus généralement les $ℝ$ -configurations test (définies comme filtrations de type fini sur l’anneau des sections), peuvent être analysées en terme de fonctions de Fubini–Study sur l’analytifié de Berkovich de $X$ pour la valuation triviale sur le corps de base. En s’appuyant sur la théorie du pluripotentiel non-archimédien, nous décrivons le complété (séparé) de l’espace des configurations test, relativement à deux pseudo-distances naturelles, en terme de fonctions plurisousharmoniques et de mesures d’énergie finie sur l’espace de Berkovich. Nous décrivons également le quotient séparé de l’espace de toutes les filtrations, et établissons une correspondance bijective entre normes divisorielles et mesures divisorielles, toutes deux déterminées par un nombre fini de valuations divisorielles.

For any polarized variety $(X, L)$ , we show that test configurations and, more generally, $ℝ$ -test configurations (defined as finitely generated filtrations of the section ring) can be analyzed in terms of Fubini–Study functions on the Berkovich analytification of $X$ with respect to the trivial absolute value on the ground field. Building on non-Archimedean pluripotential theory, we describe the (Hausdorff) completion of the space of test configurations, with respect to two natural pseudo-metrics, in terms of plurisubharmonic functions and measures of finite energy on the Berkovich space. We also describe the Hausdorff quotient of the space of all filtrations, and establish a 1–1 correspondence between divisorial norms and divisorial measures, both being determined in terms of finitely many divisorial valuations.

Reçu le : 2021-07-23
Révisé le : 2022-06-16
Accepté le : 2023-08-04
Première publication : 2024-07-03

DOI : 10.5802/aif.3668

Classification : 32Q26, 32P05
Keywords: K-stability, filtrations, valuations, test configurations, non-Archimedean geometry.
Mot clés : K-stabilité, filtrations, valuations, configurations test, géométrie non-archimédienne.

Affiliations des auteurs :

Boucksom, Sébastien ¹ ; Jonsson, Mattias ²

¹ Sorbonne Université and Université Paris Cité CNRS, IMJ-PRG F-75005 Paris (France)
² Dept of Mathematics University of Michigan Ann Arbor, MI 48109-1043 (USA)

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     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3668},
     language = {en},
     note = {Online first},
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Boucksom, Sébastien; Jonsson, Mattias. A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations. Annales de l'Institut Fourier, Online first, 99 p.

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