Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Reboulet, Rémi

doi:10.1515/crelle-2022-0059

Cited by 6 publications

(9 citation statements)

References 32 publications

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“…In the language of graded norms, the definition of appears in the work of Reboulet in a more general setting and plays a key role in his theory of geodesics in the space of non-Archimedean metrics on a line bundle [Reb22].…”

Section: Convexity and Uniquenessmentioning

confidence: 99%

The existence of the Kähler–Ricci soliton degeneration

Blum

Liu

et al. 2023

Forum of Mathematics, Pi

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Section: Convexity and Uniquenessmentioning

confidence: 99%

The existence of the Kähler–Ricci soliton degeneration

Blum

Liu

et al. 2023

Forum of Mathematics, Pi

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“…Using the results of [Reb22], one can also metrize E 1 (L an ) via setting d 1 (ϕ 0 , ϕ 1 ) = E(ϕ 0 ) + E(ϕ 1 ) − 2E(P (ϕ 0 , ϕ 1 )),…”

Section: The Non-archimedean Limitmentioning

confidence: 99%

“…6.6]. The metrization of the space E 1 0 (L an 1 ) is described in a [BJ22], but proceeds much as the metrization of E 1 (L an ) in [Reb22], while we recall that we metrize the space of maximal rays by…”

Section: The Non-archimedean Limitmentioning

confidence: 99%

The space of finite-energy metrics over a degeneration of complex manifolds

Reboulet¹

2023

Journal De L’École Polytechnique — Mathématiques

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Given a degeneration of projective complex manifolds X → D * with meromorphic singularities, and a relatively ample line bundle L on X, we study spaces of plurisubharmonic metrics on L, with particular focus on (relative) finite-energy conditions. We endow the space E 1 (L) of relatively maximal, relative finite-energy metrics with a d 1 -type distance given by the Lelong number at zero of the collection of fiberwise Darvas d 1 -distances. We show that this metric structure is complete and geodesic. Seeing X and L as schemes X K , L K over the discretely-valued field K = C((t)) of complex Laurent series, we show that the space E 1 (L an K ) of non-Archimedean finite-energy metrics over L an K embeds isometrically and geodesically into E 1 (L), and characterize its image. This generalizes previous work of Berman-Boucksom-Jonsson, treating the trivially-valued case.Résumé (L'espace des métriques d'énergie finie sur une dégénérescence de variétés complexes)Étant donné une dégénérescence de variétés projectives complexes X → D * avec des singularités méromorphes, et un fibré en droites relativement ample L sur X, nous étudions des espaces de métriques plurisousharmoniques sur L, avec une attention particulière aux conditions (relatives) d'énergie finie. Nous munissons l'espace E 1 (L) des métriques relativement maximales d'énergie finie d'une distance de type d 1 donnée par le nombre de Lelong en 0 de la famille des distances de Darvas d 1 fibre à fibre. Nous montrons que cette structure métrique est complète et géodésique. En considérant X et L comme des schémas X K , L K sur le champ discrètement valué K = C((t)) des séries de Laurent complexes, nous montrons que l'espace E 1 (L an K ) des métriques non archimédiennes d'énergie finie sur L an K s'immerge isométriquement et géodésiquement dans E 1 (L), et nous caractérisons son image. Ceci généralise un travail précédent de Berman-Boucksom-Jonsson, traitant le cas trivialement valué.

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“…Therefore, it is natural to pose the following conjecture, first explicitly stated by Reboulet [18, Theorem A], that, by the discussion above, holds in the toric case. Conjecture Let

s\mapsto h(s,\,\cdot \,)\in \mathcal {H}_A

be a subgeodesic, that is, on each chart

h=e^{-\phi}

where

\phi

is plurisubharmonic in all

n+1

variables.…”

Section: A Folklore Conjecturementioning

confidence: 99%

Chebyshev potentials, Fubini–Study metrics, and geometry of the space of Kähler metrics

Jin,

Rubinstein

2023

Bulletin of London Math Soc

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The Chebyshev potential of a Hermitian metric on an ample line bundle over a projective variety, introduced by Witt Nyström, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus‐invariant Kähler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a subgeodesic in the space of positively curved Hermitian metrics is linear in the time variable if and only if the subgeodesic is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nyström established the sufficiency. The main obstacle in the conjecture is that it is difficult to compute Chebyshev potentials, that are currently only known on the Riemann sphere and toric varieties. The goal of this article is to disprove this conjecture. To that end we characterize the geodesics consisting of Fubini–Study metrics for which the conjecture is true on the hyperplane bundle of the projective space. The proof involves explicitly solving the Monge–Ampère equation describing geodesics on the subspace of Fubini–Study metrics and computing their Chebyshev potentials.

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Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Abstract: Given a polarized projective variety ( X , L ) {(X,L)} over any non-Archimedean field, assuming continuity of envelopes, we define a metric on the… Show more

Cited by 6 publications

References 32 publications

The existence of the Kähler–Ricci soliton degeneration

The existence of the Kähler–Ricci soliton degeneration

The space of finite-energy metrics over a degeneration of complex manifolds

Chebyshev potentials, Fubini–Study metrics, and geometry of the space of Kähler metrics

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