Keywords: metrics, 2, metrics, 2, 4, nite-energy psh metrics

2.3 Pluripotential theory over nontrivially valued elds . . . . . . 2.3.1 Fubini-Study and plurisubharmonic metrics

Remerciements.

J'adresse en tout premier lieu mes profonds remerciements à mes directeurs de thèse, Catriona Maclean et Sébastien Boucksom, pour leurs qualités humaines, leur disponibilité et leur brillance qui ont permis à cette thèse de se dérouler dans des conditions idéales. Je pense garder toujours une nostalgie pour nos discussions passionnantes, à Grenoble et à Paris, sur les mathématiques et de nombreux autres sujets. I greatly thank José Ignacio Burgos Gil and David Witt Nyström for the time spent refereeing my thesis, their kind and helpful reports, and I very much look forward to discussing mathematics with both of you again.

Je remercie tous les membres du jury, notamment ceux qui n'ont pas encore été mentionnés : Antoine Ducros, Philippe Eyssidieux, Charles Favre et Emmanuel Peyre, d'avoir accepté d'en faire partie, et du temps qu'ils auront accordé à la lecture de ce manuscrit, ainsi qu'à la soutenance. Je remercie également Huayi Chen d'avoir accepté de faire partie de mon comité de suivi de thèse.

Au sein de l'Institut Fourier, c'est en premier lieu les membres des équipes administratives et du service informatique que je tiens à remercier, pour leur patience et leur gentillesse visiblement innies : à leur contact, les obligations et problèmes administratifs, de surcroît nombreux dans ce métier, se transforment en tâches agréables. Je remercie également les chercheurs permanents avec qui j'ai eu le plaisir de discuter, et desquels se dégagent, malgré l'hétérogénéité des prols, un enthousiasme et un intérêt tout à fait homogènes pour les mathématiques.

Je remercie naturellement mes amis parmi les doctorants, ex-doctorants et assimilés de l'Institut, pour tous les moments de camaraderie et de convivialité, ainsi qu'à travers diverses époques : les parties de tarot (et de télétarot), les séminaires compréhensibles, lesquels s'étendent souvent en goûter et fondue compréhensibles dont me manque la régularité d'antan ; mais aussi les discussions littéraires, musicales, et les débats absurdes pour lesquels, et contrairement à d'autres choses, nous ne semblons jamais manquer d'inspiration. Je pense plus généralement à tous mes amis : ceux rencontrés à Grenoble, mais aussi ceux dont la distance ou le Temps me sépare.

Enn, j'exprime ma reconnaissance à ma famille proche, en particulier mes parents et ma s÷ur, pour leur aide et leur soutien non seulement ces trois années durant, mais bien évidemment de plus longue date.

Les sujets perdaient en leurs discussions le relief et le poids des choses véritables, autour desquelles se concentrent les faims d'argent et de bien-être et tous les durs intérêts des hommes. Ils s'idéalisaient dans l'air léger de la raison pure, qu'aucune expérience n'épaissit. Les meilleurs n'ignoraient pas ce paradoxe. Ni Largilier, ni Augustin, ni Bruhl ne s'y méprenaient. Mais précisément cet irréel les enchantait : "Dans notre enfance, disait Largilier, nous jouions à l'explorateur et au soldat. Nous continuons de jouer."

Augustin, ou le maître est là, Joseph Malègue.

Résumé.

Nous étudions plusieurs aspects de la théorie du pluripotentiel sur un corps non-archimédien, en elle-même et à travers ses liens avec la géométrie complexe. Les objets centraux en sont les métriques plurisousharmoniques (ou psh) sur un bré en droites L au-dessus d'une variété X sur un corps nonarchimédien, dont la théorie globale a récemment été développée par Boucksom-Eriksson-Favre-Jonsson et al. Le cas le plus étudié est celui où le corps K est doté de la valeur absolue triviale ; dans cette thèse, nous nous concentrerons particulièrement sur les cas où la valeur absolue n'est pas triviale. Nous étudions dans un premier temps l'image de l'opérateur de Fubini-Study asymptotique sur des corps non-archimédiens généraux, qui permet d'approcher des métriques plurisousharmoniques sur un bré en droites ample à l'aide de normes agissant sur les sections de ses puissances. Ensuite, nous construisons des géodésiques plurisousharmoniques dans les espaces de métriques non-archimédiennes psh d'énergie nie sur un bré ample, et étudions leurs propriétés de régularité, ce qui étend des constructions classiques du monde complexe au monde non-archimédien. Enn, nous considérons une dégénérescence analytique de variétés complexes X sur le disque unité, que nous identions avec une variété X K sur le corps non-archimédien K des séries de Laurent à coecients complexes. Etant donné un bré en droites relativement ample L sur X, nous construisons l'espace métrique géodésique des métriques relativement maximales d'énergie nie sur L. Nous montrons que l'espace des métriques d'énergie nie non-archimédiennes sur L K (ayant également identié L avec une variété sur le corps K) se plonge isométriquement et géodésiquement dans le précédent, ce qui permet de déduire la convexité d'incarnations non-archimédiennes de diverses fonctionnelles en lien avec la K-stabilité.

7

Abstract.

We study and develop pluripotential theory over a non-Archimedean eld, in itself, and through its interactions with complex geometry. Its main objects are plurisubharmonic (or psh) metrics on a line bundle L over a variety X over a non-Archimedean eld. The global theory of such psh metrics has recently been developed by Boucksom-Eriksson-Favre-Jonsson et al. The most well known case is that of a eld K endowed with the trivial absolute value; in this thesis, we will focus on elds endowed with nontrivial absolute values. We rst look into the image of the asymptotic Fubini-Study operator over general non-Archimedean elds, which allows us to approximate plurisubharmonic (psh) metrics on an ample line bundle L using norms acting on the sections of the tensor powers of L. Then, extending an important construction from complex geometry to the non-Archimedean world, we show that there exist plurisubharmonic geodesics in spaces of nite-energy psh metrics on an ample line bundle, and study their regularity properties with respect to the regularity of their endpoints. Finally, we consider an analytic degeneration of complex varieties, X, bred over the unit disc, which we identify with a variety X K over the non-Archimedean eld of complex Laurent series. Given a relatively ample line bundle on X, we construct the geodesic metric space of relatively maximal nite-energy metrics on L. We show that the space of non-Archimedean nite-energy metrics on L K (having once again identied L with a variety over the eld K) embeds isometrically and geodesically into the former, allowing us to deduce convexity of some non-Archimedean versions of various functionals related to K-stability. 8 Contents Remerciements.

Résumé.

Abstract.

Introduction en français.

Introduction.

1 Algebraic preliminaries.

Introduction en français.

Le sujet d'étude principal de cette thèse est la théorie du pluripotentiel nonarchimédien, dont les objets centraux sont les métriques (pluri)sousharmoniques non-archimédiennes. Nous expliquerons bientôt plus en détail ces termes. Nous développons cette théorie en suivant deux axes.

Premièrement, en elle-même: ses fondements sont récents ( [START_REF] Boucksom | Solution to a non-Archimedean Monge-Ampère equation[END_REF], [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], [BE]), et particulièrement dans le cas de valuation non triviale, beaucoup d'aspects demeurent mystérieux. Il est donc de mise d'essayer de transposer divers concepts et énoncés du monde complexe au monde non-archimédien, ce qui est parfois très délicat. Dans plusieurs cas, le comportement de certains objets diverge complètement du cas complexe, ce qui donne de nouvelles perspectives intéressantes.

Ensuite, nous l'étudions à travers ses applications à la géométrie complexe, puisque dans certains cas, la géométrie non-archimédienne encode de manière très ecace des phénomènes singuliers ou asymptotiques liés à des objets complexes. Dans ce cadre, nous nous spécialiserons au cas où le corps de base est soit le corps C muni de la valeur absolue triviale, ou le corps C((t)) des séries formelles à coecients complexes, muni de sa valeur absolue tadique usuelle.

De la géométrie complexe à la géométrie non-Archimédienne.

Le principe GAGA pour les variétés algébriques sur un corps non-archimédien.

Soit X une variété algébrique projective sur C. Le célèbre théorème "GAGA" de Serre ( [SerGAGA]) arme que l'on peut lui associer une variété analytique projective complexe, X an , en en préservant les propriétés topologiques essentielles (connexité, séparation, compacité), ainsi que les données des faisceaux cohérents. Cela permet d'importer des techniques analytiques puissantes, venues de l'analyse complexe et de la géométrie diérentielle, an d'étudier des problèmes de nature algébrique.

Soit maintenant X une variété algébrique projective sur un corps K nonarchimédien, c'est-à-dire un corps muni d'une valeur absolue |•|, complet pour la topologie induite par celle-ci, et dont l'inégalité triangulaire est ranée en l'inégalité ultramétrique: pour tous x, y dans K, |x + y| ≤ max(|x|, |y|).

Des exemples fondamentaux de tels corps sont:

1. C((t)), le corps des séries de Laurent formelles sur C, muni de la valeur absolue t-adique;

2. pour p premier, Q p , le corps des p-adiques, muni de la valeur absolue p-adique;

3. pour p premier, F p ((t)), le corps des séries de Laurent formelles à coecients dans le corps ni F p , que l'on munit également de la valeur absolue t-adique;

4. tout corps K peut être muni de la valeur absolue triviale, égale à 1 sur K × , qui est non-archimédienne.

Ces exemples mettent en valeur la diversité et l'utilité du monde non-archimédien.

Les corps p-adiques sont omniprésents en théorie des nombres (on notera des résultats classiques comme le principe de Hasse-Minkowski, le lemme d'Hensel, le théorème de Mahler, mais aussi des applications aux équations diérentielles p-adiques, et à la théorie du potentiel p-adique) et leurs extensions perfectoïdes ont donné naissance à une théorie fertile ( [Scholze], [FF] ). Le corps F p ((t)) permet de dénir des voisinages formels de courbes arithmétiques. La classe des variétés sur le corps C((t)), nous y reviendrons longuement au cours de cette thèse, contient comme cas particulier les dégénérescences de variétés analytiques complexes. Le cas de valuation triviale enn, malgré sa dénomination, a un fort intérêt géométrique, sur lequel nous travaillerons également.

Il paraît souhaitable d'avoir à disposition un arsenal analytique similaire à celui du monde complexe pour les K-variétés. Cependant, des dicultés se manifestent rapidement. En premier lieu, l'inégalité ultramétrique a des conséquences notables sur la topologie de K : celle-ci est totalement discontinue, c'est-à-dire que ses seuls connexes sont les points et l'ensemble vide. Se pose ensuite le problème de dénir une notion de variété analytique, et donc de fonction analytique, sur K. La manière naïve de procéder consiste à imiter le cas complexe, en dénissant une fonction analytique sur K d comme une série entière convergente à coecients dans K, puis de procéder par recollements.

Le résultat est catastrophique : Serre ([Ser65]) montre que toute variété compacte en ce sens est isomorphe à une union disjointe de boules {|x| ≤ 1}. Il n'est pas acceptable qu'un tel objet puisse être l'analytication d'une variété algébrique sur K!

Nous observons à travers ces phénomènes, comme l'a fait Berkovich en son temps, qu'il convient de "rajouter des points" à un candidat pour la putative analytication X an de X. Nous ne décrirons pas la procédure en détail dans cette introduction ; brièvement, l'analytié dit de Berkovich X an s'identie à la compactication d'un ensemble de valuations d'origine géométrique. Les points cruciaux de cette construction sont les suivants :

En particulier, si le corps K de base est de valuation triviale, alors X an s'identie à l'ensemble des valuations sur les corps de fractions K(Y ), pour toutes les sous-variétés irréductibles Y ⊂ X.

Théorie du pluripotentiel géométrique.

Le point de départ est le suivant. Etant donné un domaine Ω ⊂ C d , une fonction f lisse (ou simplement deux fois dérivable) à valeurs réelles sur Ω est dite plurisousharmonique si sa hessienne complexe est une matrice positive. Cette dénition peut être comprise comme généralisant la notion de fonction convexe (lisse ou deux fois dérivable) de variables réelles, et de telles fonctions jouissent de propriétés similaires.

Plus généralement, une fonction f semi-continue supérieurement et localement intégrable sur Ω est dite plurisousharmonique si elle peut être écrite comme limite décroissante d'un let de fonctions lisses plurisousharmoniques ; ou, de manière équivalente, si sa hessienne complexe au sens des distributions est une mesure positive.

De telles fonctions admettent une généralisation naturelle aux variétés complexes, et aux brés en droites sur celles-ci. Commençons par supposer qu'X est une variété projective complexe compacte, de dimension d, que l'on munit d'un bré en droites L ample. Qu'L soit ample équivaut à demander que, pour tous entiers k assez grand, le produit tensoriel L ⊗k (que l'on notera additivement par kL) dispose d'une base (s i ) dim H 0 (X,kL) i=1

de l'espace H 0 (X, kL)

de ses sections, tels que l'application

x → [s 1 (x), . . . , s dim H 0 (X,kL) (x)] soit un plongement dans l'espace projectif P dim H 0 (X,kL)-1 .

Soit τ Ω une trivialisation L| Ω C × Ω de L. Une métrique singulière sur L est une métrique φ qui s'écrit, pour toutes telles trivialisations et pour ( , x) ∈ C × Ω, |τ Ω ( )| e -φ Ω (x) = , où les poids locaux φ Ω appartiennent à L 1 loc (Ω). Le courant de courbure de L, qui est donné par c 1 (L) = dd c φ (qui est bien déni en vertu de la condition d'intégrabilité locale) et est en fait indépendant de φ, est positif (au sens français) si et seulement si les poids locaux φ Ω sont tous plurisousharmoniques, auquel cas l'on dira que la métrique est elle-même plurisousharmonique, et l'on écrira φ ∈ PSH(X, L).

Grâce aux résultats de Demailly, nous pouvons en fait caractériser la classe des métriques psh sur L comme la plus petite classe de métriques singulières sur L qui:

1. est stable par maxima nis; 2. est stable par limites décroissantes de lets; 3. est stable par addition de constantes réelles; 4. contient toutes les métriques de type Fubini-Study, c'est-à-dire les métriques de la forme

φ = 1 2k log i |s i | 2 e 2λ i ,
où les s i forment une base de sections d'une puissance kL, sans pointbase, et les λ i sont des constantes réelles.

Supposons maintenant qu'X est une variété sur un corps non-archimédien K.

Sur un espace de Berkovich, il n'y a pas de calcul diérentiel à proprement parler. Nous ne pouvons donc pas utiliser de caractérisations locales pour dénir des métriques plurisousharmoniques sur L an , comme nous l'avons fait dans le cas complexe.

Il faut donc être plus astucieux: nous allons imiter a posteriori comme dénition d'une métrique psh non-archimédienne, l'énoncé du résultat précédent de Demailly. Le seul ingrédient manquant dans le cas non-archimédien est la notion de métrique Fubini-Study. An d'avoir les mêmes propriétés, il convient de remplacer les sommes de carrés par des maxima (un thème récurrent, comme nous le verrons).

Denition 0.0.0.1. Soit K un corps non-archimédien, X une K-variété projective compacte et L un bré en droites ample sur X. Une métrique φ sur L an est dite de Fubini-Study si elle peut s'écrire de la forme

φ = 1 k log max i |s i |e λ i ,
où les s i forment une base de sections d'une puissance kL, sans point-base, et les λ i sont des constantes réelles. Nous noterons FS(L an ) cette classe.

Denition 0.0.0.2. Avec les conventions de la précédente dénition, une métrique φ sur L an est dite plurisousharmonique si elle peut s'écrire comme la limite décroissante d'un let de métriques Fubini-Study sur L an . Nous noterons PSH(L an ) cette classe.

Il est alors possible de voir qu'est vraie la même caractérisation que dans le cas complexe: la classe des métriques psh sur L an est la plus petite classe de métriques singulières sur L an qui:

1. est stable par maxima nis;

2. est stable par limites décroissantes de lets;

3. est stable par addition de constantes réelles;

4. contient toutes les métriques de type Fubini-Study sur L an .

Métriques d'énergie nie.

Un objet fondamental dans l'étude des métriques psh sur un bré en droites holomorphe est l'opérateur de Monge-Ampère, qui associe à une métrique psh lisse φ le produit (dd c φ) d . Dans le cas où φ n'est plus lisse (ou du moins C 2 ), dd c φ est strictement un courant et non plus une forme diérentielle : leur produit n'est plus déni. Le travail remarquable de Bedford-Taylor ( [BT]) a permis d'étendre cet opérateur (ainsi que des produits plus généraux de la forme (dd c φ 1 ) ∧ • • • ∧ (dd c φ d )) à la classe des métriques localement bornées.

Cela empêche toutefois de considérer des métriques singulières ; cependant, des décennies plus tard, cet opérateur sera étendu par Boucksom-Eyssidieux-Guedj-Zeriahi ( [BEGZ]) à la classe E 1 (X, L) de métriques d'énergie nie, que nous dénissons maintenant.

Posons deux métriques φ 0 et φ 1 , lisses, et dénissons leur énergie de Monge-Ampère ainsi :

E(φ 0 , φ 1 ) = (d + 1) -1 d i=0 (φ 0 -φ 1 ) (dd c φ 0 ) i ∧ (dd c φ 1 ) d-i .

Il est remarquable qu'ayant xé la métrique φ 1 , l'expression

φ → E(φ, φ 1 )

est décroissante en φ. Par le théorème de régularisation de Demailly ([Dem92], [START_REF] Blocki | On regularization of plurisubharmonic functions on manifolds[END_REF]), toute métrique psh sur L peut être réalisée comme une suite décroissante de métriques psh lisses, et l'on peut dénir une extension de E à PSH(X, L) en posant E(φ, φ 1 ) = lim k→∞ E(φ k , φ 1 ) pour φ ∈ PSH(X, L), et k → φ k une suite de métriques psh lisses sur L décroissant vers φ. Notons que cette énergie peut alors prendre la valeur -∞.

Nous pouvons ainsi dénir l'espace des métriques plurisousharmoniques d'énergie nie sur L:

E 1 (X, L) = {φ ∈ PSH(X, L), E(φ, φ 1 ) > -∞, ∀ φ 1 ∈ C ∞ ∩ PSH(X, L)}.
Notons que E satisfait la propriété de cocycle

E(φ 0 , φ 1 ) = E(φ 0 , φ 2 ) + E(φ 2 , φ 1 ),
comme il peut se voir sur des métriques lisses, puis par régularisation ; ce qui implique que la classe E 1 est indépendante du choix de métrique de référence, ce qui est mis en évidence par notre notation.

La classe E 1 contient strictement la classe des métriques localement bornées : en particulier, elle contient "beaucoup" de métriques psh singulières. C'est une classe qui est intrinsèquement intéressante, de surcroît grâce à [BEGZ], où l'opérateur de Monge-Ampère y est comme promis étendu.

Il a été observé par Darvas ([Dar17]) que cette classe pouvait être dotée d'une structure d'espace métrique complet, comme suit : l'on dénit l'enveloppe "toit" (rooftop) de deux métriques φ 0 , φ 1 dans E 1 (X, L) comme la régularisation semi-continue supérieurement

P (φ 0 , φ 1 ) = usc sup{ψ ∈ PSH(X, L), ψ ≤ φ 0 , φ 1 }.
C'est la généralisation de l'enveloppe convexe de deux fonctions convexes ; si les deux métriques sont d'énergie nie, alors leur enveloppe toit est également d'énergie nie, et l'on écrit d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 )).

Il est également possible de dénir les opérateurs de Monge-Ampère associés à des métriques continues psh non-archimédiennes, par la théorie de l'intersection (suivant Gubler, Boucksom-Eriksson-Favre) ou bien la théorie des formes diérentielles sur les espaces de Berkovich (d'après Chambert-Loir et Ducros, se basant sur les superformes de Lagerberg).

Cela nous permet de dénir l'énergie de Monge-Ampère en imitant exactement le cas complexe, ainsi que la classe

E 1 (L an )
des métriques psh d'énergie nie sur L an . De la même manière, nous pouvons introduire la distance d 1 (comme fait dans [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF]) sur E 1 (L an ), en utilisant les enveloppes psh non-archimédiennes. Nous nous heurtons ici à un premier problème : celui de la continuité des enveloppes, ]), puis dans [DZ], [CSW], [BBJ], [Zha]. Résumé des résultats obtenus.

Les travaux de

Quantication de métriques non-Archimédiennes par des espaces de normes.

Depuis les travaux de Bouche-Tian-Catlin-Zelditch, il est connu que l'on peut approximer uniformément une métrique continue psh φ (complexe) par des métriques Fubini-Study φ k associées à φ.

Plus précisément, on peut construire pour tout k une norme hermitienne ζ k sur l'espace des sections H 0 (kL + K X ), donnée par

ζ k (s) 2 = ˆX |s| 2 e -kφ :
en eet, |s| 2 e -kφ dénit bien une mesure sur X.

ζ k+ (s k • s ) ≤ ζ k (s k ) • ζ (s ).
A chacune de ces normes nous associons une métrique de Fubini-Study

FS k (ζ k ) = 1 k log sup s∈H 0 (kL) |s|/ζ k (s).
FS • (ζ • ) = usc (lim k FS k (ζ k )).
L'opérateur FS • , que nous appellerons opérateur de Fubini-Study asymptotique, n'est pas injectif sur l'espace des normes graduées bornées sur L.

Cependant, il devient injectif si l'on prend le quotient cet espace par une relation d'équivalence algébrique, que l'on décrit maintenant.

Tout espace de normes N (V ) sur un K-espace vectoriel V de dimension nie peut être muni d'une distance d 1 , modélisée sur la distance d 1 dans les espaces euclidiens. Si L possède de bonnes propriétés de positivité, alors la limite des distances d 1,k sur les N (H 0 (kL)) "converge" en une semi-distance sur l'espace des normes graduées bornées sur L,

d 1 (ζ • , ζ • ) := lim k d 1,k (ζ k , ζ k )
existe, et satisfait l'inégalité triangulaire et la symétrie. Notons pour simplier N (L) l'espace quotient pour la relation

d 1 (ζ • , ζ • ) = 0.
Nous prouvons alors au Chapitre 3 : Theorem 0.0.0.3. L'opérateur de Fubini-Study asymptotique réalise un bijection entre son image PSH ↑ (L an ) et l'espace N (L), qui est de surcroît une isométrie.

Ici, les distances considérées sont la distance de Darvas héritée de E 1 (L an ) ⊃ PSH ↑ (L an ) à gauche, et la distance quotient de d 1 à droite. L'image PSH ↑ (L an ) peut être caractérisée comme l'ensemble des métriques psh sur L an approchables par en-dessous, soit les métriques qui sont limites croissantes de lets de métriques psh.

Ce résultat est la version non-trivialement valuée d'un résultat de Boucksom-Jonsson dans [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF]. Nous nous appuyons fondamentalement sur l'article [START_REF] Chen | Distribution of logarithmic spectra of the equilibrium energy[END_REF], en en utilisant les résultats concernant l'asymptotique de fonctions superadditives sur les corps d'Okounkov. Nous nous basons également sur le formalisme développé dans [BE].

Géodésiques et segments maximaux dans les espaces de métriques non-Archimédiennes.

Dans [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], ici le Chapitre 4, nous nous intéressons à la transposition dans le monde non-Archimédien d'objets essentiels en géométrie complexe : les géodésiques dans les espaces de métriques plurisousharmoniques.

Commençons par décrire la situation dans le monde complexe pour un instant. C'est encore une fois la recherche de métriques Kähler-Einstein qui est à l'origine de l'intérêt de la construction de géodésiques dans l'espace H(L) des métriques lisses et strictement psh sur L.

A = {e -1 ≤ |z| ≤ 1}
en posant t = -log |z|. Alors, pour toute métrique psh A z → ψ z invariante par rotation sur L × A, que l'on peut également voir comme un segment psh t → ψ t , si l'on a ψ 0 ≤ φ 0 et ψ 1 ≤ φ 1 , alors pour tout t dans [0, 1],

ψ t ≤ φ t .
Cette propriété caractérise les segments géodésiques de Mabuchi. Nous pouvons donc les réaliser comme l'enveloppe

sup{ψ segment psh, ψ 0 ≤ φ 0 , ψ 1 ≤ φ 1 }.
Il se trouve que cette dénition a du sens plus généralement si les métriques au bord ne sont plus continues : c'est l'approche utilisée par Darvas pour parler de géodésiques faibles dans E 1 (L). Notons qu'une caractérisation alternative est que φ est un segment psh le long duquel l'énergie de Monge-Ampère est ane.

Supposons maintenant que (X, L) est une variété compacte polarisée sur un corps K non-archimédien. Evidemment, il n'est plus possible de dénir une notion de géodésique dans PSH(L an ) au sens riemannien. Cependant, nous pouvons dénir une classe de segments Fubini-Study assez naturelle, qui sont des maxima nis de segments de la forme Theorem 0.0.0.4. Etant donné φ 0 , φ 1 ∈ E 1 (L), nous posons d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 ))

[0, 1] t → 1 k max i (log |s i | + tλ i + (1 -t)λ i ),
. Alors, 1. On dénit premièrement une notion de métrique psh adaptée à ce contexte : on requiert que nos métriques admettent une extension psh à un certain modèle analytique (X , L) de (X, L). Theorem 0.0.0.5. L'espace ( Ê1 (L), d1 ) est un espace métrique géodésique et complet. Il admet de surcroît des segments d1 -geodésiques maximaux parmi tous les segments géodésiques, qui peuvent être construits comme familles de segments géodésiques bre à bre.

Ensuite, à toute métrique φ NA d'énergie nie dans PSH(L), on associe une métrique non-Archimédienne sur l'analytié L an de L vu comme une variété sur C((t)). Sur un point divisoriel de X an , qui est associé à un diviseur D dans la bre centrale d'un modèle de X, la valeur de φ NA correspond au nombre de Lelong générique de φ le long de D. Elle est étendue de manière unique par une propriété générale des fonctions psh non-Archimédiennes: elles sont dénies de manière unique par leurs valeurs aux points divisoriels.

On étudie ensuite l'évolution de l'énergie de Monge-Ampère le long d'une métrique φ ∈ Ê1 (L), particulièrement quand z → 0. Plus précisément, on étudie la métrique

E(φ) : z → φ d+1 z = E(φ z )
sur le bré L sur le disque épointé, qui peut être vue comme une famille d'énergies de Monge-Ampère intrinsèques. On montre que, dans le cas où φ s'étend de manière localement bornée sur un modèle analytique de (X, L), on a :

(E(φ)) NA = E NA (φ NA ),
ce qui peut être interprété comme disant que le nombre de Lelong des énergies bre à bre coïncide avec l'énergie de Monge-Ampère non-archimédienne de φ NA . Cela généralise le cas classique, puisqu'un nombre de Lelong n'est qu'une pente à l'inni généralisée.

Cette égalité n'est pas vraie dans le cas général. La classe de métriques de

Ê1 (L) satisfaisant à (E(φ)) NA = E NA (φ NA ),
est dénie comme la classe des métriques hybridement maximales Ê1 hyb (L), et généralise la classe des rayons géodésiques maximaux de [BBJ]. On obtient une caractérisation extrémale de ces métriques, et on montre que cette classe est isométrique à la classe des métriques non-Archimédiennes E 1 (L an ).

On obtient comme conséquence un résultat heuristique sur la convexité de fonctionnelles non-Archimédiennes.

Theorem 0.0.0.6.

• Il y a un plongement isométrique de

(E 1 (L an ), d NA 1 )
dans ( Ê1 (L), d1 ) avec pour image Ê1 hyb (L).

• Un segment psh dans Les Chapitres 3, 4, et 5 consistent essentiellement des résultats de (respectivement) [START_REF] Reboulet | The asymptotic Fubini-Study operator over general non-Archimedean elds[END_REF], [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], et [START_REF] Reboulet | The space of nite-energy metrics over a degeneration of complex manifolds[END_REF], comme expliqué précédemment.

Introduction.

The main subject of this thesis is non-Archimedean pluripotential theory, in which the role of central object is played by non-Archimedean (pluri)subharmonic metrics. Naturally, we will explain those terms in due time. We shall develop this theory along two axes.

We shall rst study it in and for itself: the foundations are recent ([BFJ15], [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], [BE]), and many aspects remain mysterious, especially in the case of nontrivial valuation. It seems therefore appropriate to try and transpose various concepts and results from the complex to the non-Archimedean world. This is often more delicate than it seems. In many cases, the behaviour of some objects is completely dierent from the complex case, opening interesting directions of research.

Secondly, we will study non-Archimedean pluripotential theory through its applications to complex geometry. Indeed, in certain cases, non-Archimedean geometry is a very ecient way to encode singular or asymptotic behaviours of complex objects. For this approach, we will specialize to the case where the base eld is either C together with its trivial absolute value, or the eld C((t)) of complex Laurent series together with its t-adic absolute value.

From complex geometry to non-Archimedean geometry.

The GAGA principle for varieties over a non-Archimedean eld.

Let X be a projective algebraic variety over the eld C. The now-famous GAGA principle of Serre ( [SerGAGA]) asserts that one can associate to such X a projective complex manifold, X an , preserving its essential topological properties (such as connectedness, separatedness, and compactness), as well as data given by coherent sheaves (in the sense that there is an equivalence of categories between the categories of coherent sheaves on X and X an ). This allows one to use powerful analytic methods coming from complex analysis and dierential geometry in order to study problems of algebraic nature.

Consider now a projective algebraic variety X over a non-Archimedean eld ). The eld F p ((t)) is used to dene formal neighbourhoods of arithmetic curves. The class of varieties over C((t)), as we shall explain in much detail over the course of this manuscript, contains as a particular case degenerations of complex manifolds. Finally, the case of trivial valuation holds much geometric importance, in spite of its name; we will work on this side of the story as well.

It therefore seems appropriate to have at hand analytic tools the likes of which are available in the complex world. Many diculties arise shortly. First of all, the ultrametric inequality has notable consequences on the topology of K, for it is totally discontinuous (i.e. its only connected sets are the null sets and singletons). One also needs to dene a notion of an analytic space (and also of analytic functions) over K. Naively, one would mimick the complex case, by dening an analytic function on K d to be a converging power series with coecients in K; then, to dene analytic spaces via gluing. The result is rather catastrophic: Serre ([Ser65]) shows that any compact analytic space in this sense is a disjoint union of unit balls; a notably poor candidate for the analytication of a K-algebraic variety!

As Berkovich in his time, we then notice that our putative analytication X an of X needs "additional points". Although we will not explain the exact procedure here, we shall describe it briey: X an is roughly identied with the compactication of valuations of geometric origin on K(X). The points of utmost importance are as follows:

1. the analytication preserves topological properties such as connectedness, separatedness, and compactness;

2. it realizes an equivalence of categories between the categories of coherent sheaves on X and X an ; 3. it is also dened for varieties over Archimedean elds -in particular C, where it coincides with the usual analytication in the sense of Serre.

In particular, if the base eld K is trivially valued, then X an corresponds to the space of valuations on elds of fractions K(Y ) for all irreducible subvarieties Y ⊂ X, endowed with the topology of pointwise convergence.

Geometric pluripotential theory.

Given a domain Ω ⊂ C d , a smooth (or twice dierentiable) function f : Ω → R is plurisubharmonic if its complex Hessian is a positive matrix. One can see this denition to generalize the notion of a convex (smooth or twice dierentiable) function, and indeed plurisubharmonic functions share many properties with convex functions.

More generally, an upper semicontinuous and locally integrable real-valued function f on Ω is plurisubharmonic if it can be realized as a decreasing limit of a net of smooth plurisubharmonic metrics ; or equivalently, if its complex Hessian in the sense of distributions denes a positive measure.

The denition of such functions naturally extends to complex varieties, and to line bundles on them. Let us begin by xing a complex compact projective manifold X, of dimension d, endowed with an ample line bundle L. We recall that L is ample if and only if, for all large enough integers k, the k-fold tensor product L ⊗k (denoted additively kL) admits a basis of sections (s i )

dim H 0 (X,kL) i=1
such that the map

x → [s 1 (x), . . . , s dim H 0 (X,kL) (x)]

denes an embedding in projective space P dim H 0 (X,kL)-1 .

Let τ Ω be a trivialization L| Ω C × Ω of L. A singular metric on L is a metric φ on L which can be written, in all such trivializations and for all ( , x)

∈ C × Ω, as |τ Ω ( )| e -φ Ω (x) = ,
where the local weights φ Ω belong to L 1 loc (Ω). The curvature current of L, given by c 1 (L) = dd c φ (which one notices to be well-dened due to the local integrability condition) turns out to be independent of φ, and is (frenchly) positive if and only if the local weights φ Ω are all plurisubharmonic, in which case we shall say that φ is itself plurisubharmonic, and we shall write φ ∈ PSH(X, L).

Due to results of Demailly, we may in fact characterize the class of plurisubharmonic (psh) metrics on L as the smallest class of singular metrics on L which:

1. is closed under nite maxima; 2. is closed under decreasing limits of nets; 3. is closed under addition of real constants; 4. contains all Fubini-Study metrics, i.e. metrics written as

φ = 1 2k log i |s i | 2 e 2λ i ,
where the s i are a basepoint-free basis of sections of kL, and the λ i are real constants.

Let us now assume X to be a variety over a non-Archimedean eld K. On a Berkovich space, there is no notion of dierential calculus. We thus cannot use local characterizations as before to dene plurisubharmonic functions on X an and metrics on L an , as is done in the complex case.

We will thus mimick a posteriori the statement of Demailly's result to take as our denition of a non-Archimedean psh metric. The only missing ingredient remains the notion of a Fubini-Study metric. As we will notice to be a recurrent theme, we simply "replace sums of squares with maxima".

Denition 0.0.0.7. Let K be a non-Archimedean eld, X a compact projective K-variety, and L an ample line bundle on X. A metric φ on L an is Fubini-Study provided one can write it as

φ = 1 k log max i |s i |e λ i ,
where the s i are a basepoint-free basis of some power kL, and the λ i are real constants. We will denote this class of metrics by FS(L an ).

Denition 0.0.0.8. With the conventions of the previous Denition, a metric φ on L an is plurisubharmonic if it can be written as the decreasing limit of a net of Fubini-Study metrics on L an . We will denote this class of metrics by PSH(L an ).

One can then see that the same characterization of the class of psh metrics holds as in the complex case: it is the smallest class of singular metrics on L an which:

1. is closed under nite maxima;

2. is closed under decreasing limits of nets;

3. is closed under addition of real constants;

4. contains all Fubini-Study metrics on L an .

Finite-energy metrics.

A fundamental object in the study of psh metrics on a holomorphic line bundle is the Monge-Ampère operator, associating to a smooth metric φ the product (dd c φ) d . In the case where φ is no longer smooth (or twice dierentiable), dd c φ is strictly a current and no longer a dierential form: such a product is therefore no longer dened. The remarkable work of Bedford-Taylor ( [BT]) allowed to extend this operator (as well as more general products of the form

(dd c φ 1 ) ∧ • • • ∧ (dd c φ d ))
to the class of locally bounded metrics. This still prevents us from considering possibly singular metrics; however, decades later, this operator will be extended by the work of Boucksom-Eyssidieux-Guedj-Zeriahi ( [BEGZ]) to the class E 1 (X, L) of nite-energy metrics, which we describe now.

Let us x two smooth psh metrics φ 0 and φ 1 , and let us dene their Monge-Ampère energy as follows:

E(φ 0 , φ 1 ) = (d + 1) -1 d i=0 (φ 0 -φ 1 ) (dd c φ 0 ) i ∧ (dd c φ 1 ) d-i .
Remarkably, having xed the right-hand metric φ 1 , the expression

φ → E(φ, φ 1 )
is decreasing in φ. By the Demailly regularization Theorem ([Dem92], [START_REF] Blocki | On regularization of plurisubharmonic functions on manifolds[END_REF]), any psh metric on L can be realized as a decreasing limit of smooth psh metrics, so that one can extend E to PSH(X, L) by setting

E(φ, φ 1 ) = lim k→∞ E(φ k , φ 1 )
for φ ∈ PSH(X, L), and k → φ k a sequence of smooth psh metrics on L decreasing to φ. Let us note that this energy can possibly take the value -∞.

We therefore dene the space of nite-energy metrics on L:

E 1 (X, L) = {φ ∈ PSH(X, L), E(φ, φ 1 ) > -∞, ∀ φ 1 ∈ C ∞ ∩ PSH(X, L)}.
Note that E satises the cocycle property:

E(φ 0 , φ 1 ) = E(φ 0 , φ 2 ) + E(φ 2 , φ 1 ),
as one can see on smooth metrics, then by regularization; this implies the class E 1 to be independent of the choice of a reference metric.

Darvas [START_REF] Darvas | The Mabuchi completion of the space of Kähler potentials[END_REF]) later observed that this class could be endowed with the structure of a complete metric space, as follows: we dene the "rooftop" envelope of two metrics φ 0 , φ 1 in E 1 (X, L) as the usc regularization

P (φ 0 , φ 1 ) = usc sup{ψ ∈ PSH(X, L), ψ ≤ φ 0 , φ 1 }.
This generalizes the convex envelope of two convex functions; if both metrics have nite energy, then their envelope also has nite energy, and we write d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 )).

One may also dene Monge-Ampère operators associated to continuous non-Archimedean psh metrics, using either intersection theory (Gubler, Boucksom-Eriksson-Favre) or dierential forms on Berkovich spaces (Chambert-Loir and Ducros, building on the superforms of Lagerberg).

By mimicking exactly the denition in the complex case, we may then dene the Monge-Ampère energy as well as the class

E 1 (L an )
of nite-energy psh metrics on L an . Similarly, we may introduce the distance d 1 (as in [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF]) on E 1 (L an ), using non-Archimedean psh envelopes.

We run into a rst problem: that of continuity of envelopes, that is: the fact that the psh envelope of a continuous metric is itself continuous. This is a property of the pair (X, L), classically true in the complex case if X is (say) normal and L is ample; in the non-Archimedean case, this property is partially conjectural, under similar hypotheses. It is currently known in all cases useful for applications to complex geometry, i.e. when the base eld is either C with its trivial absolute value, of C((t)) with its natural absolute value. We review the currently known cases in the Section in Chapter 2 dedicated to non-Archimedean psh envelopes.

Concluding our preliminaries, we briey describe a recent application of non-Archimedean pluripotential theory to complex geometry, after Berman, Boucksom, and Jonsson: a variational approach to the Yau-Tian-Donaldson conjecture.

The existence of constant curvature metrics (in all possible avours) has been a recurring theme in dierential geometry for many decades. A problem of particular interest is that of existence of Kähler-Einstein metrics on a compact Kähler manifold X, i.e. a smooth strictly psh metric φ on the canonical bundle K X , whose associated (1, 1)-form ω is proportional to its Ricci curvature Ric(ω). The existence of such a metric is a very strong condition, subordinate to topological conditions: the class c 1 (X) itself has to be signed.

By the works of Aubin and Yau, it is known that if X is canonically polarized (with ample canonical bundle) or Calabi-Yau (with trivial canonical bundle), there exists a Kähler-Einstein metric on X, which is furthermore unique in the former case.

The Fano case (with antiample canonical bundle) is notoriously more dicult.

Inspired by various ideas (from GIT quotient theory, in particular the Kempf-Ness Theorem and its consequences in symplectic geometry; as well as the then recent result of Donaldson-Uhlenbeck-Yau proving equivalence between an algebraic notion of stability for holomorphic vector bundles and the existence of Hermitian Yang-Mills connections on them), Donaldson formulates a conjecture, further and further rened, known by the name of Yau-Tian-Donaldson conjecture, according to which existence of a Kähler-Einstein metric on X is equivalent to the positivity of some purely algebraic quantities (so-called Futaki invariants). It has now been proven several times, rst by Chen-Donaldson-Sun ([CDS1], [START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π[END_REF], [START_REF] Chen | Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof[END_REF]), then in [DZ], [CSW], [BBJ], [Zha].

The works of Berman-Boucksom-Jonsson ([BBJ]), some parts of which we generalize over the course of this thesis, give a non-Archimedean formulation of the aforementioned conjecture ([BBJ, Theorem A, Corollary 5.2]): existence of a Kähler-Einstein metric on X is equivalent to coercivity of some functional (the Ding functional) on the space of non-Archimedean psh metrics on K an X , the analytication of K X with respect to the trivial absolute value on C. Generalizations of this conjecture have also been studied, most fundamentally the existence of cscK metrics which has recently been reduced by Li ( [Li]) to a conjecture purely in the domain of non-Archimedean pluripotential theory; and the more general case of solitons has similarly been studied by ).

Summary of the main results.

Quantization of non-Archimedean metrics via spaces of norms.

Since the works of Bouche-Tian-Catlin-Zelditch, it is known that a (complex) continuous psh metric φ can uniformly be approximated by Fubini-Study metrics φ k associated to φ.

Precisely, one can construct for all k a Hermitian norm ζ k on the space of sections H 0 (kL + K X ), given by

ζ k (s) 2 = ˆX |s| 2 e -kφ :
indeed, |s| 2 e -kφ does dene a measure on X. Picking a basis (s j ) j of this space of sections, orthonormal for ζ k , one denes its Bergman kernel

B φ,k := j |s j | 2 .
This kernel is in fact independent of the choice of such a basis, and the metric

φ k = 1 k log B φ k
is Fubini-Study. We then have that the φ k converge uniformly to the original metric, φ.

The purpose of the article [START_REF] Reboulet | The asymptotic Fubini-Study operator over general non-Archimedean elds[END_REF] is to study problems of similar nature in the non-Archimedean setting. Using a "quantization" procedure as above, we will also characterize a class of non-Archimedean psh metrics using purely algebraic data.

Let us x a non-Archimedean, nontrivially valued eld K. Instead of having as initial data a continuous metric on L an , we shall rather consider a sequence ζ • = (ζ k ) k of ultrametric norms on each graded piece H 0 (kL) of the section algebra of L, further satisfying a compatibility condition with respect to multiplication of sections: it must be submultiplicative, i.e. for any two sections s k ∈ H 0 (kL), s ∈ H 0 ( L), we have that

ζ k+ (s k • s ) ≤ ζ k (s k ) • ζ (s ).
To each such norm we associate a Fubini-Study metric by setting

FS k (ζ k ) = 1 k log sup s∈H 0 (kL) |s|/ζ k (s).
Assuming our graded norm to satisfy an additional growth condition (which we shall not make explicit here, but roughly ensuring that it does not "blow up"), in which case we shall speak of a bounded graded norm, the submultiplicativity condition implies via Fekete's Lemma that the limit

lim k FS k (ζ k )
exists. Its usc regularization is then a non-Archimedean psh metric which we denote

FS • (ζ • ) = usc (lim k FS k (ζ k )).
The operator FS • so dened, which we will call the asymptotic Fubini-Study operator, is not in general injective on the space of bounded graded norms on L. It however becomes injective by taking the quotient of this space under some algebraic equivalence relation, which we describe now.

Any space of norms N (V ) on a K-vector space v of nite dimension can be endowed with a distance, d 1 , modelled on the distance d 1 in Euclidean spaces.

If L has sucient positivity properties, the limit of the distances d 1,k on each N (H 0 (kL)) "converges" to a pseudodistance on the space of bounded graded norms on L, in the sense that for any two such graded norms ζ • , ζ • , the limit

d 1 (ζ • , ζ • ) := lim k d 1,k (ζ k , ζ k )
exists, and satises the symmetry property as well as the triangle inequality.

Let us write for clarity N (L) for the quotient space under the relation

d 1 (ζ • , ζ • ) = 0.
In Chapter 3, we prove the following:

Theorem 0.0.0.9. The asymptotic Fubini-Study operator realizes a bijection between its image PSH ↑ (L an ) and the space N (L), which is furthermore an isometry.

Here, the distances in question in the isometry statement are the Darvas distance inherited from E 1 (L an ) ⊃ PSH ↑ (L an ) on the left, and the quotient d 1 distance on the right. The image PSH ↑ (L an ) can be characterized as the set of psh metrics on L an approximable by below, i.e. metrics that are limits of increasing nets of psh metrics.

This result is the nontrivially-valued version of a result of Boucksom-Jonsson in [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF]. We extensively use the results of [START_REF] Chen | Distribution of logarithmic spectra of the equilibrium energy[END_REF]concerning asymptotics of superadditive functions on Okounkov bodies. We also build on the formalism developed in [BE].

Geodesics in spaces of non-Archimedean metrics.

In [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], or in this thesis Chapter 4, we transpose essential objects from the complex to the non-Archimedean world: geodesics in spaces of plurisubharmonic metrics.

Let us begin with the complex case. The purpose of such geodesics is again sparked by the search for Kähler-Einstein metrics in the space H(K X ) of smooth and strictly psh metrics on the canonical bundle. The Kähler-Einstein equation admits a certain functional, the K-energy of Mabuchi, as its Euler-Lagrange equation; in other words, critical points of the K-energy are solutions to the Kähler-Einstein problem. In order to continue the variational approach, it would be desirable that K be convex along certain distinguished segments of metrics in H(K X ). The ideal candidates would be ane lines, for H is a convex set, but they do not satisfy that property. Instead, K is convex with respect to another Riemannian structure on H, by the work of Mabuchi ([Mab]), whose geodesics are given by a Monge-Ampère equation (see Semmes ([Sem]), Donaldson ([Don99])). X.X. Chen ([CX00]) was the rst to show existence of such geodesics in H.

Such geodesics are characterized by their maximality, as we explain. Pick two smooth, strictly psh metrics φ 0 , φ 1 on K X , and denote by [0, 1] t → φ t the Mabuchi geodesic joining them. One can see it as a rotation-invariant psh metric on the product K X × A → X × A, where A is the annulus

A = {e -1 ≤ |z| ≤ 1},
upon setting t = -log |z|. We then have that, for any rotation-invariant psh metric A z → ψ z on K X × A (which we can similarly see as a psh segment t → ψ t ), if ψ 0 ≤ φ 0 and ψ 1 ≤ φ 1 , then for all t in [0, 1],

ψ t ≤ φ t .
This property completely characterizes Mabuchi geodesic segments, which are for this reason sometimes called "maximal psh segments". We can furthermore realize them as the envelope

sup{ψ segment psh, ψ 0 ≤ φ 0 , ψ 1 ≤ φ 1 }.
As it turns out, this denition also makes sense more generally on any line bundle L on X, and also if the endpoint metrics are no longer continuous: this is the approach of Darvas to dene weak geodesics in E 1 (L). Finally, a last characterization of such geodesics is that φ is a psh segment along which the Monge-Ampère energy is ane.

Let us now assume (X, L) to be a polarized variety over a non-Archimedean eld K. Obviously, geodesics in PSH(L an ) in the Riemannian sense can no longer be dened. However, we may still dene a rather natural class of Fubini-Study segments, as nite maxima of segments of the form

[0, 1] t → 1 k max i (log |s i | + tλ i + (1 -t)λ i ),
where (s i ) i is again a basepoint-free basis of some H 0 (kL), and the λ i , λ i are real constants. One then denes psh segments as decreasing limits of nets of Fubini-Study segments. Then, inspired by the extremal characterization in the complex case, we develop in [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF] (Chapter 4 of this manuscript) a theory for geodesics in spaces of non-Archimedean nite-energy metrics. We rst treat the case where the endpoint metrics are continuous, where we dene our geodesic as an envelope as above, and we show via a Legendre transform that such a geodesic remains continuous in time and space.

Then, adapting the quantization approach of Darvas-Lu-Rubinstein in our context, we show that the Monge-Ampère energy is ane along such geodesics, and that they do dene metric geodesics with respect to the non-Archimedean d 1 distance. On the algebraic side, we construct geodesics between bounded graded norms.

Finally, we extend our construction to the case of nite-energy endpoints, via decreasing approximations. Our results may be summarized as follows:

Theorem 0.0.0.10. Given φ 0 , φ 1 ∈ E 1 (L), and setting

d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 )),
we have that:

1. (E 1 (L), d 1
) is a metric space; 2. there exists a unique maximal psh segment t → φ t joining φ 0 and φ 1 ; 3.

φ t ∈ E 1 (L) for all t ∈ [0, 1];
4. the segment φ t is a metric geodesic with respect to the d 1 distance dened above, i.e. there exists c ≥ 0 with

d 1 (φ t , φ s ) = c • |t -s|
for all t, s ∈ [0, 1]; 5. the Monge-Ampère energy is ane along the segment φ t , and any psh segment satisfying such a property with endpoints φ 0 and φ 1 has to coincide with the geodesic segment joining φ 0 et φ 1 . If the endpoint metrics φ 0 and φ 1 are furthermore continuous, then the maximal psh segment joining them is also continuous in time and space.

Spaces of metrics on degenerations of complex manifolds.

Many recent works over the course of the previous decade, notably by Berman-Boucksom-Favre-Hisamoto-Jonsson et al., have been dedicated to the study of non-Archimedean limits of plurisubharmonic geodesic rays. Precisely, given a compact projective complex manifold X with an ample line bundle L, a geodesic ray

[0, ∞) t → φ t ∈ PSH(X, L)
is identied with a rotation-invariant psh metric Φ on the pullback of L to the trivial product D * × X, where the variable z given by the rst projection is identied with t via t = -log |z|. The limit as t → ∞ denes, on studying the singularities of Φ along certain compactications of the product, a non-Archimedean metric on the analytication of X with respect to the trivial absolute value on C, as explained before.

The purpose of [START_REF] Reboulet | The space of nite-energy metrics over a degeneration of complex manifolds[END_REF], Chapter 5 in this thesis, is to generalize such considerations to the case of an arbitrary degeneration π : X → D * of complex projective manifolds without necessarily assuming S 1 -invariance, or even isotriviality. Non-Archimedean phenomena immediately arise: one can realize such a degeneration as a variety over the eld C((t)).

We rst dene a notion of a (family of) psh metrics adapted to this context.

Let L be a relatively ample line bundle on X. We denote by PSH(L) the space of psh metrics which can be extended to some analytic model (X , L) of (X, L), i.e. to a polarized complex analytic space (X , L) bred over the unpunctured unit disc, which is isomorphic to (X, L) outside of its bre over zero. Generalizing the work of Berman-Darvas-Lu ( [BDL]), who dene a radial d 1 distance on the space of nite-energy psh rays via

d 1 (φ, ψ) = lim t d 1 (φ t , ψ t ) t
(the d 1 in the right-hand side being the "classical" Darvas distances), we show that taking a generalized Lelong number of the map dened on D * of brewise d 1 distances also denes a distance on the space of "relatively maximal" brewise nite-energy metrics in PSH(L). The notion of relative maximality essentially states that they are maximal in the usual pluripotential theory sense over all preimages of relatively compact open sets in the punctured disc. We will write Ê1 (L) for this class of metrics, and d1 for this distance.

We then show the following:

Theorem 0.0.0.11. The space ( Ê1 (L), d1 ) is a complete, geodesic metric space. There furthermore exist distinguished d1 -geodesic segments, maximal among all geodesic segments, which can be constructed as brewise families of geodesic segments in each E 1 (L z ).

Then, to any brewise nite-energy metric φ in PSH(L), we associate a non-Archimedean metric φ NA on the analytication L an of L seen as a variety over C((t)). On a divisorial point of X an , associated to a divisor D inside the central bre of some analytic model of X, the value of φ NA roughly corresponds to the generic Lelong number of φ along D.

We then study the singular limit of the brewise Monge-Ampère energy along a brewise nite-energy metric in PSH(L), i.e. the metric

E(φ) : z → φ d+1 z = E(φ z )
on the Deligne pairing bundle L over the punctured disc. We show that, if φ extends as a locally bounded metric to some analytic model of (X, L),

then (E(φ)) NA = E NA (φ NA ),
i.e. the Lelong number of the brewise Monge-Ampère energies corresponds with the non-Archimedean Monge-Ampère energy of φ NA .

Such an equality does not hold in general. The class of metrics inside Ê1 (L)

satisfying

(E(φ)) NA = E NA (φ NA ),
is what we dene to be the class of "hybrid maximal" metrics Ê1 hyb (L). One can see this to generalize the maximal geodesic rays of Berman-Boucksom-Jonsson, as a Lelong number is merely a generalized slope at innity. We then obtain an extremal characterization of such metrics, and show that this class is isometric to the class E 1 (L an ): we have realized non-Archimedean metrics as purely complex geometric objects. As a consequence, we obtain a heuristic result regarding proving convexity of non-Archimedean energy functionals. Our second main result is the following:

Theorem 0.0.0.12.

• There exists an isometric embedding

(E 1 (L an ), d NA 1 )
in ( Ê1 (L), d1 ) with image Ê1 hyb (L).

• A psh segment Ê1 hyb (L) is geodesic if and only if its image E 1 (L an K ) is a non-Archimedean geodesic in the sense of Chapter 4.

• One has a general "plurifunctional extension property": given d + 1 relatively ample line bundles L i on X, for all (d + 1)-uple of metrics

φ i ∈ Ê1 hyb (L i ), we have ( φ 0 , . . . , φ d X/D * ) NA = φ NA 0 , . . . , φ NA d .
Organization of the manuscript.

In Chapter 1, we develop algebraic preliminaries necessary for the main results of this manuscript. We begin with generalities on non-Archimedean elds. Then, given such a eld K, we study metric and spectral properties of spaces of norms on nite-dimensional K-vector spaces. Finally, we extend this study to the case of graded norms on K-graded algebras generated in degree one. This last part contains some results from [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], as well as an unpublished result regarding completeness of a certain space of graded norms with respect to a d ∞ -type distance.

In Chapter 2, we look into geometric preliminaries. After briey recalling Berkovich's construction, we focus on various types of metrics on analytications of K-line bundles, and explain basic notions of non-Archimedean pluripotential theory.

Finally, Chapters 3, 4, et 5 essentially contain the results of respectively [START_REF] Reboulet | The asymptotic Fubini-Study operator over general non-Archimedean elds[END_REF], [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], and [START_REF] Reboulet | The space of nite-energy metrics over a degeneration of complex manifolds[END_REF], as previously explained.

Chapter 1

Algebraic preliminaries.

As a convention throughout this manuscript, rings and elds shall be assumed to be commutative.

1.1 Non-Archimedean elds.

(i) |0| = 0, |1| = 1; (ii) |xy| = |x| • |y|, for x, y ∈ R; (iii) |x + y| ≤ |x| + |y|, for x, y ∈ R.
It is furthermore ultrametric or non-Archimedean if the triangle inequality (iii) is rened to the ultrametric inequality:

(iiib) |x + y| ≤ max{|x|, |y|}, for x, y ∈ R.
If R is a eld, in which case we will usually denote it by K, a multiplicative seminorm on K will be referred to as an absolute value on K, and an ultrametric multiplicative seminorm as an ultrametric or a non-Archimedean absolute value on K. A eld K endowed with an absolute value shall be referred to as a valued eld.

Example 1.1.1.2. An example to keep in mind is the trivial absolute value, dened as

|x| 0 = 1, x = 0,
which is non-Archimedean, and well-dened for any eld K, by the eld axioms.

Remark 1.1.1.3. To any absolute value | • | on a eld K is associated its

valuation ν(•) := -log | • |.
To make sense of the following denition, note that an absolute value on a eld K always endows it with a topology, induced by the distance d(x, y) := |x -y|, for x, y ∈ K.

Denition 1.1.1.4. We will dene a non-Archimedean eld to be a eld K which is Cauchy-complete with respect to the topology induced by a non-Archimedean absolute value | • |. We dene an Archimedean eld to be a eld K which is Cauchy-complete with respect to the topology induced by an absolute value | • | which is not non-Archimedean.

1.1.2 Classication of complete valued elds.

For the remainder of this Section, we shall x a valued eld (K, | • |).

As we see from the classical Gelfand-Mazur Theorem, the classication of Archimedean elds is very simple.

Theorem 1.1.2.1 ([AC5-7, VI, $6, Théorème 1]). If (K, |•|) is an Archimedean eld, then it is isomorphic to either R or C endowed with their usual Archimedean absolute values.

The non-Archimedean case has more structure: they are classied by bicharacteristic and by their value group, as we now explain. We assume for now that K is non-Archimedean.

Consider the valuation ν(•) = -log | • | associated to the absolute value on K.

By the absolute value axioms, its possible values dene an additive subgroup of R, the value group of (K, |•|). It is a well-known fact that such subgroups are either trivial, nontrivial discrete, or dense.

Denition 1.1.2.2. We say that K is:

• trivially valued if its value group is trivial (equivalently if it is endowed with the trivial absolute value);

• discretely valued if its value group is nontrivial and discrete;

• densely valued if its value group is dense.

We now look into the second way to classify non-Archimedean elds: the pair given by their characteristic, and the characteristic of their residue eld.

Denition 1.1.2.3. To any non-Archimedean eld K, one may associate the following objects:

• its valuation ring K • , dened as the set of elements in K with absolute value ≤ 1;

• the maximal ideal of K • , K •• , characterized as the set of elements in K • (or K) with absolute value < 1; • its residue eld K = K • /K •• .
Only three cases are possible, by which we say that:

• K has equicharacteristic 0 if K and K are of characteristic 0;

• K has mixed chracteristic if K is of characteristic 0 and K is of characteristic p, p prime;

• K has equicharacteristic p if K and K are of characteristic p, p prime. Remark 1.1.2.4. One can see that the valuation ring of a non-Archimedean eld is Noetherian if and only if the eld is discretely valued.

We are now equipped to understand the main examples of non-Archimedean elds.

Example 1.1.2.5. We rst look at the discretely-valued case.

• the eld C((t)) of complex formal Laurent series is discretely valued, and has equicharacteristic 0, for its residue eld is C;

• the eld Q p of p-adics, p prime, is discretely valued and has mixed characteristic, for its residue eld is F p ;

• the eld F p ((t)) of formal Laurent series over the nite eld F p , p prime, is discretely valued, and has equicharacteristic p, for its residue eld is F p .

The densely valued world is "wilder". We nally mention a last classication result, of potential interest to the reader.

Theorem 1.1.2.6 (Cohen's structure Theorem, [SerBook, $4, Théorème 2]).

If K is an equicharacteristic discretely valued non-Archimedean eld, then it is isomorphic to the eld K((t)), the eld of formal Laurent series over its residue eld.

Maximal completeness.

We briey review a property for non-Archimedean eld which will be essential in our study of norms on K-vector spaces later on. For this section, our references are [START_REF] Poonen | Maximally complete elds[END_REF] and [START_REF] Barría | Summary on non-Archimedean valued elds[END_REF].

Denition 1.

1.3.1. Let (K, | • |) be a non-Archimedean eld. Let (L, | • | ) be a valued eld extension of (K, | • |).
We say that it is an immediate extension if the value groups of K and L are isomorphic, and if K and L are isomorphic.

Denition 1.1.3.2. A non-Archimedean eld which admits no immediate extension will be called a maximally complete eld.

Example 1.1.3.3. Any equicharacteristic discretely valued eld is maximally complete. The eld Q p , p prime, is also maximally complete.

Our claim is then the following:

Proposition 1.1.3.4. Any nontrivially valued non-Archimedean eld can be embedded into a eld which is: 1. densely valued; 2. Cauchy-complete; 3. algebraically closed; 4. maximally complete; 5. of the same bicharacteristic as the original eld. Proof. Let K be as in the above statement. Let K be the Cauchy completion of the algebraic closure of K. Taking algebraic closures in the discretely or densely valued case already yields a densely valued eld. K still has the same bicharacteristic as K. Then, [Kap42, Theorem 5] ensures that there exists a maximally complete immediate extension of K, which then has the desired properties by denition.

Example 1.1.3.5. Let us begin with a eld of complex formal Laurent series C((t)). The eld of formal Puiseux series is an algebraic closure thereof, but as mentioned before, it is not Cauchy-complete. The Levi-Civita eld L[Q, C] of formal complex power series P with supp(P ) ∪ (-∞, k] nite for all k ∈ Z is then a Cauchy closure of the eld of complex Puiseux series. Its maximally complete immediate extension is given by the eld of Hahn series C((Q)) of formal complex power series P with supp(P ) well-ordered in Q. If one were to start from F p instead, one would have to rst pass to an algebraic closure of F p , and proceed as before ([CS18, Theorem 7.2]). If one started from Q p , one would obtain a p-adic Mal'cev-Neumann eld as in [Poo93, Section 4].

Spaces of norms on vector spaces over non-Archimedean elds.

Throughout this Section, unless otherwise specied, we x a non-Archimedean eld K, with absolute value | • |; and a nite-dimensional vector space V over K, of dimension d.

Spaces of norms.

Denition 1.2.1.1. A norm on V is a function

ζ : V → R + ,
satisfying the following properties:

• ζ(v) = 0 if and only if v = 0 V ; • ζ(λ • v) = |λ| • ζ(v), for λ ∈ K, v ∈ V ; • ζ(v + w) ≤ max{ζ(v), ζ(w)}, for v, w ∈ V .
We denote by

N (V )
the set of norms on V . This space is closed under the (pointwise) maximum operation, which we denote

ζ ∨ ζ = max(ζ, ζ ), for any two norms ζ, ζ ∈ N (V ). Denition 1.2.1.2. A norm ζ ∈ N (V ) is diagonalizable if there exists a basis (e 1 , . . . , e d ) of V such that, for all v = v i e i , with v i ∈ K for all i, we have that ζ(v) = max i |v i | • ζ(e i ).
We also say that this basis is orthogonal for ζ. We say that it is a lattice norm, or a pure diagonalizable norm, if, for all i, ζ(e i ) = 1. We dene

N diag (V ), N latt (V )
as respectively the set of diagonalizable norms and the set of lattice norms on V .

Remark 1.2.1.3. In our conventions, we will dene a lattice of V to be

a submodule L of V of nite type over K • , such that L ⊗ K o K = V .
In particular, the unit ball of a lattice norm is always a lattice, justifying the terminology.

Remark 1.2.1.4. It is also common practice to use the terminology of cartesian bases, see e.g. [BGR, Ch. 2].

We note that, when K is maximally complete, we know from [START_REF] Bosch | Non-Archimedean analysis[END_REF]2.4.4] (see also [BE,Lemma 1.12]) that any norm ζ on V admits an orthogonal basis. Further emphasizing the dierence between the possible value groups, we have:

Lemma 1.2.1.5 ([BE, L1.29]). Let K be nontrivially valued.

• if K is discretely valued, the unit ball of any diagonalizable norm is a lattice of V ;

• if K is densely valued, the unit ball of a norm ζ ∈ N (V ) is a lattice if and only if ζ is a lattice norm.
There are many ways to construct norms from pre-existing ones. The two building blocks are the following:

• let W ⊂ V be a subspace; any norm ζ induces a quotient norm ζ V /W on V /W , as follows: given [v] ∈ V /W , ζ V /W ([v]) = inf w∈W ζ(v + w);
• a norm ζ on V induces a norm ζ ⊗n on any tensor power V ⊗n of V , by setting, for each v ∈ V ⊗n ,

ζ ⊗n (v) = inf v= i v i 1 ⊗•••⊗v i n max i (ζ(v i 1 ) × • • • × ζ(v i n )),
where we take the inmum over all possible decompositions of v of the form

i v i 1 ⊗ • • • ⊗ v i n ,
where the sum over i is nite, and v i k ∈ V for all i, k.

We may combine these constructions. Let n be an integer, λ be a partition of n, and let S λ denote the Schur functor associated to λ. Then a norm ζ ∈ N (V ) denes a norm ζ λ ∈ N (S λ (V )), as this vector space is a composition of quotients of tensor products. In particular, ζ ∈ N (V ) induces:

• a norm ζ ∧n on the n-fold exterior product V ∧n ;

• a norm ζ n on the n-fold symmetric product V n .

Relative spectra.

Generalizing the set of eigenvalues of the transition matrix between to jointly orthonormalized complex Hermitian norms, we introduce the relative spec-

λ i (ζ, ζ ) = sup W ∈ i≤r≤dim V Gr K (r,V ) inf w∈W -{0} [log ζ (w) -log ζ(w)],
where

Gr K (r, V ) denotes the r-th Grassmannian of V . Remark 1.2.2.2. When K is maximally complete, again from [BGR, 2.4.4],
for any two norms, there exists a basis diagonalizing both of them (we shall in that case speak of codiagonalizing bases).

By [BE, P2.24], if both norms are diagonalizable by a basis (s i ), ordered such

that i > j ⇒ ζ (s i ) ζ(s i ) ≥ ζ (s j ) ζ(s j ) , then λ i (ζ, ζ ) = log ζ (s i ) -log ζ(s i ).
Due to the diagonalizability issues explained before in non-maximally complete elds, it will be desirable to pass to a maximally complete eld extension, such as one of the lovely elds promised by Proposition 1.1.3.4. We now see how to extend norms, and how spectra behave under this operation.

V L = V ⊗ K L is dened as ζ L (v ) = inf max i |a i | • ζ(v i ),
for any v ∈ V L , where the inf is dened over all representations

v = i a i • v i , with coecients a i in L and v i ∈ V .
σ(ζ, ζ ) = d -1 δ λ i (ζ,ζ ) ,
where we recall that

d = dim K V . Denition 1.2.3.2. Let p ∈ [1, ∞).
d p (ζ, ζ ) p = ˆR |λ| p dσ(ζ, ζ ).
We furthermore dene

d ∞ (ζ, ζ ) = max λ∈Sp(ζ,ζ )
|λ|.

Of utmost interest for our considerations are the cases d 1 and d ∞ , which have more practical expressions:

d ∞ (ζ, ζ ) = sup v∈V -{0} | log ζ (v) -log ζ(v)|,
and

d 1 (ζ, ζ ) = d -1 λ i (ζ, ζ ), where d = dim K V .
v ∈ V , e -C ζ (v) ≤ ζ(v) ≤ e C ζ (v).
vol(ζ, ζ ) = d 1 (ζ, ζ ),
and, reversing the inequality, we obtain

-vol(ζ, ζ ) = d 1 (ζ, ζ ).
Theorem 1.2.3.6 ([BE, T2.25]). Let {e 1 , . . . , e d } be a basis of V . We have that

vol(ζ, ζ ) = 1 d log ζ ∧d (e 1 ∧ • • • ∧ e d ) -log ζ ∧d (e 1 ∧ • • • ∧ e d )
Corollary 1.2.3.7. Volumes satisfy a cocycle property: given a third norm

ζ ∈ N (V ), vol(ζ, ζ ) = vol(ζ, ζ ) + vol(ζ , ζ ).
Proposition 1.2.3.8 ([BE, P1.8, T1.19, L1.29]). With respect to the distance d ∞ on N (V ), we have that:

(i) N (V ) is complete; (ii) N diag (V ) is dense in N (V ), with equality if K is discretely valued; (iii) if K is discretely valued, N latt (V ) is discrete and closed in N (V ); (iv) if K is densely valued, N latt (V ) is dense in N diag (V ).
We note that relative volumes behave well with respect to the d ∞ distance.

Lemma 1.2.3.9 (Volumes are Lipschitz, [BE, P2.14]). The mapping vol is 1-Lipschitz in both variables, and thus Lipschitz on the product N (V )×N (V ).

In other words, given two pairs of norms

(ζ 0 , ζ 1 ) and (ζ 0 , ζ 1 ) acting on V , we have | vol(ζ 0 , ζ 1 ) -vol(ζ 0 , ζ 1 )| ≤ d ∞ (ζ 0 , ζ 0 ) + d ∞ (ζ 1 , ζ 1 ).
Finally, we see that taking quotients of norms is a contracting operation for the d ∞ distance.

Proposition 1.2.3.10 (Quotients decrease distance). Let W ⊂ V be a proper linear subspace of V , let ζ and ζ be norms on V . Denote ζ and ζ the induced quotient norms on V /W . We then have that:

d ∞ ( ζ, ζ ) ≤ d ∞ (ζ, ζ ). Proof. Fix a = d ∞ (ζ, ζ ), and ṽ ∈ V /W . It is enough to show that e -a ζ (ṽ) ≤ ζ(ṽ) ≤ e a ζ(ṽ).
We lift ṽ to a sum v + w with v ∈ V -W and w ∈ W . Note that

e -a ζ (v + w) ≤ ζ(v + w),
for all such lifts, so that we can pass to the inf and get that e -a ζ (ṽ) ≤ ζ(ṽ).

Similarly, we get that e -a ζ(ṽ) ≤ ζ (ṽ).

The result follows.

1.2.4 Apartments.

Norms oftentimes like to be roommates. This anity gives insight into the building-like structure of the space of diagonalizable norms on the vector space V . Pick a basis s = (s i ) i of V , and the projection

ι s : R d → N (V ) = N diag (V )
dened by sending a vector α = (α 1 , . . . , α d ) to the unique norm ζ diagonalized in the basis s, and with

ζ(s i ) = e -α i
for all i. The image of this injection map is called the apartment A s associated to the basis s. It inherits the geometry of R d , in the sense that for any distance d p , p ∈ [1, ∞], ι S realizes an isometry onto its image for the distances d p as dened in the previous section.

The space of diagonalizable norms N diag (V ), as the (non-disjoint!) union of all the apartments s basis of V A s then inherits a complete Euclidean building structure for p = 2, in the sense of [START_REF] Remy | Bruhat-Tits buildings and analytic geometry[END_REF]. As two diagonalizable norms may be diagonalized in the same basis, it follows that any pair of norms share some apartment in the building

N diag (V ). If K is maximally complete, all norms are diagonalizable, so that N diag (V ) = N (V ).
i ) = ζ 0 (s i ) 1-t • ζ 1 (s i ) t .
From an elementary computation it follows that it is geodesic for all distances d p .

Remark 1.2.5.1. In the case p > 1, the norm geodesic is the only geodesic segment between two norms. However, if p = 1, there are innitely many geodesic segments between two norms, reecting the d 1 geometry of R d .

Consider now two non-necessarily diagonalizable norms ζ 0 and ζ 1 ∈ N (V ), and approximations by diagonalizable norms

ζ n 0 , ζ n 1 such that d 1 (ζ n i , ζ i ) ≤ 1 n for i = 0, 1. Dene ζ t = lim n ζ n t ,
where t → ζ n t is the norm geodesic dened above joining ζ n 0 and ζ n 1 . We make the following claim: Proposition 1.2.5.2. The limit ζ t above exists, and satises the following properties:

1. it is independent of the approximation; 2. it is controlled uniformly by the endpoints: given approximations (ζ m i ) m , i = 0, 1, of the bounds, we have

d 1 (ζ t , ζ m t ) ≤ (1 -t)d 1 (ζ 0 , ζ m 0 ) + td 1 (ζ 1 , ζ m 1 );
3. ζ t is the pointwise limit of the approximating norms: for s ∈ V ,

ζ t (s) = lim m ζ m t (s).
This claim relies on the diagonalizable case of Corollary 1.2.5.8, which has yet to be proven. However, all the results in this Section hold in the general non-necessarily diagonalizable case. In order to avoid writing each statement twice, we rst prove Proposition 1.2.5.2 using the diagonalizable case of Corollary 1.2.5.8, and then state each result of this Section in the general case. Circular reasoning is avoided by following the logical order "proof of a result in the diagonalizable case" → "proof of Corollary 1.2.5.8 and Proposition 1.2.5.2" → "proof of a result in the non-diagonalizable case".

Proof. Pick m, n ≥ 0, and write using Corollary 1.2.5.8:

d 1 (ζ m t , ζ n t ) ≤ (1 -t)d 1 (ζ m 0 , ζ n 0 ) + td 1 (ζ m 1 , ζ n 1 )
which establishes that the sequence (ζ n t ) is d 1 -Cauchy, thus has a limit in N (V ). This also establishes the statement about uniform approximation by passing to the limit in n in the previous inequality. Pick now a second pair of approximations (ζ n 0 ), (ζ n 1 ), and write ζ n t for the norm geodesic between the adequate bounds for all n. Then,

d 1 (ζ n t , ζ n t ) ≤ (1 -t)d 1 (ζ n 0 , ζ n 0 ) + td 1 (ζ n 1 ,
ζ t (s) ≤ ζ 0 (s) 1-t ζ 1 (s) t .
Proof. Let (s i ) be a basis codiagonalizing the endpoints, and write s as a i •

s i , so that ζ t (s) = max i |a i | • ζ t (s i ) = max i |a i | • ζ 0 (s i ) 1-t ζ 1 (s i ) t = max i |a i | 1-t ζ 0 (s i ) 1-t • |a i | t ζ 1 (s i ) t ≤ (max i |a i | • ζ 0 (s i )) 1-t • (max i |a i | • ζ 1 (s i )) t = ζ 0 (s) 1-t ζ 1 (s) t .
The non-diagonalizable case follows upon approximation with diagonalizable norms and passing to the pointwise limit in the log-convexity inequalities

ζ m t (s) ≤ ζ m 0 (s) 1-t ζ m 1 (s) t .
We now prove an important comparison inequality concerning norm geodesics with comparable endpoints. They will be crucial in proving many later results, including the metric convexity of d 1 in spaces of norms. Proof. Assume rst all norms to be diagonalizable. Write then a section s of inH 0 (kL) as s = a i s i where (s i ) is a basis codiagonalizing ζ 0 and ζ 1 (hence all the ζ t ), so that

ζ t (s) ≤ max i |a i | • ζ t (s i ) ≤ max i |a i | • ζ 0 (s i ) 1-t ζ 1 (s i ) t ≤ max i |a i | • ζ 0 (s i ) 1-t ζ 1 (s i ) t = ζ t (s),
where we have used the ultrametric inequality, log-convexity of ζ t , the inequalities in the hypotheses, then the denition of a basis diagonalizing ζ t . This concludes the proof. If we do not have diagonalizability, one uses approximations by diagonalizable norms and Proposition 1.2.5.2(iii) to conclude.

As explained before, one can take determinants of norms on V . We now look into the behaviour of geodesics under this operation, which will allow us to prove metric convexity of geodesics. We rst recall the following result from [BE]: Lemma 1.2.5.5. Let ζ be a norm on a d-dimensional K-vector space V , and

let s = (s i ) i be a basis of V . If s diagonalizes ζ, then det ζ(s 1 ∧ • • • ∧ s d ) = d i=1 ζ(s i ).
We recall that the determinant of a norm

ζ on V is the norm induced by ζ on det V = V ∧d . Lemma 1.2.5.6. Let t → ζ t be a norm geodesic in V . Then, t → det ζ t is the one-dimensional norm geodesic joining det ζ 0 and det ζ 1 in det V .
Proof. By density, we may assume that there exists a basis s = (s i ) i of V diagonalizing the ζ t for all t. We have:

det ζ t (s 1 ∧ • • • ∧ s d ) = ζ t (s i ) = ζ 0 (s i ) 1-t ζ 1 (s i ) t = (det ζ 0 (s 1 ∧ • • • ∧ s d )) 1-t (det ζ 1 (s 1 ∧ • • • ∧ s d )) t ,
which by denition proves the statement. We have used Lemma 1.2.5.5

for the rst equality, the diagonalizing property of (s i ) for the second, and Lemma 1.2.5.5 again for the third.

This yields a proof that the relative volume of geodesics is ane:

Corollary 1.2.5.7. Given two norm geodesics

t → ζ t , t → ζ t in V , the function vol(ζ t , ζ t ) = t → log det ζ t det ζ t is ane.
Proof. By density again, without loss of generality we can assume that we can pick a basis s = (s i ) i diagonalizing ζ t for all t. We then have that for all t,

det ζ t (s 1 ∧ • • • ∧ s d ) = d i=1 ζ t (s i ) = d i=1 ζ 0 (s i ) 1-t ζ 1 (s i ) t = (det ζ 0 (s 1 ∧ • • • ∧ s d )) 1-t (det ζ 1 (s 1 ∧ • • • ∧ s d )) t . Furthermore,
log det ζ t det ζ t ≤ log (det ζ 0 (s 1 ∧ • • • ∧ s d )) 1-t (det ζ 1 (s 1 ∧ • • • ∧ s d )) t (det ζ 0 (s 1 ∧ • • • ∧ s d )) 1-t (det ζ 1 (s 1 ∧ • • • ∧ s d ))
d 1 (ζ t , ζ t ) ≤ (1 -t)d 1 (ζ 0 , ζ 0 ) + td 1 (ζ 1 , ζ 1 ).
Proof.

d 1 (ζ t , ζ t ) ≤ (1 -t) vol(ζ 0 , ζ 0 ∨ ζ 0 ) + t vol(ζ 1 , ζ 1 ∨ ζ 1 ) + (1 -t) vol(ζ 0 , ζ 0 ∨ ζ 0 ) + t vol(ζ 1 , ζ 1 ∨ ζ 1 ) = (1 -t)d 1 (ζ 0 , ζ 0 ) + td 1 (ζ 1 , ζ 1 ),
proving the general statement.

Remark 1.2.5.9. From the general theory of metric spaces, a result such as Corollary 1.2.5.8 ensures existence and good properties of the "cone at innity" or boundary at innity of N (V ), which can be described as equivalence classes of (norm) geodesic rays in N (V ) staying at bounded distance.

The same result holds in d ∞ distance:

Lemma 1.2.5.10 (Convexity of d ∞ along norm geodesics). Let ζ 0 , ζ 1 and ζ 0 ,

ζ 1 be four norms on V , and denote by ζ t , resp. ζ t the norm geodesic joining the rst two, resp. the last two. Then,

d ∞ (ζ t , ζ t ) ≤ (1 -t)d ∞ (ζ 0 , ζ 0 ) + td ∞ (ζ 1 , ζ 1 ).
Proof. As usual, the general case follows from the diagonalizable case by approximation, therefore we make this assumption. We have that

d ∞ (ζ t , ζ t ) = max i |λ i (ζ t , ζ t )| = max i |(1 -t)λ i (ζ 0 , ζ 0 ) + tλ i (ζ 1 , ζ 1 )| ≤ (1 -t) max i |λ i (ζ 0 , ζ 0 )| + t max i |λ i (ζ 1 , ζ 1 )|,
and the last term is simply

(1 -t)d ∞ (ζ 0 , ζ 0 ) + td ∞ (ζ 1 , ζ 1 ), which is the desired result.
1.3 Spaces of norms on graded algebras over non-Archimedean elds.

Throughout this Section, (K, | • |) will be a non-Archimedean eld, and V • will denote a graded K-algebra

V • = k∈N V k , such that V 0 = K, each V k is a nite-dimensional K-vector space.
We assume that V • is furthermore generated in degree one in that the multiplication morphisms

V k 1 → V k
(where denotes the symmetric product) are surjective for all k ∈ N * .

1.3.1 Bounded graded norms.

An algebra norm on V • compatible with the grading may be characterized as the data of norms ζ • = (ζ k ) k acting on each V k , satisfying the following

submultiplicativity condition: given v k ∈ V k and v ∈ V , we must have that ζ k+ (v k • v ) ≤ ζ k (v k ) • ζ (v ).
A sequence of norms on V • satisfying this condition is called a graded norm.

In order to study asymptotic properties of graded norms, using e.g. Fekete's lemma, we will need a growth condition that is both natural and ensures that the graded norms does not "blow up". We explain here how to formulate such a condition algebraically. This requires an additional denition.

Denition 1.3.1.1. We shall say that a graded norm

ζ • on V • is generated in degree one if, for all k ∈ N * , ζ k is the quotient norm induced by the surjective symmetry morphism V k 1 → V k .
Example 1.3.1.2. If K is trivially valued, the most simple example of a graded norm generated in degree one is the trivial graded norm ζ triv,• ,

given by ζ triv,k (v k ) = 1 for all v k ∈ V k -{0 V k }.
, ζ • ∈ N • (V • ).
Then, the sequence of rescaled measures

(k • dim K (V k )) -1 * σ(ζ k , ζ k )
weakly converges to a compactly supported probability measure. We call this limit measure the relative spectral measure of ζ • and ζ • :

σ(ζ • , ζ • ) = lim k (k • dim K (V k )) -1 * σ(ζ k , ζ k ).
Remark 1.3.2.2. This Theorem is proven in [CM15, Theorem 5.2], see also [BE,Theorem 9.5] and the similar statement in [START_REF] Reboulet | The asymptotic Fubini-Study operator over general non-Archimedean elds[END_REF], building on ideas of [START_REF] Witt | Transforming metrics on a line bundle to the Okounkov body[END_REF]. The trivially-valued case is proven in [BJ18a, Theorem 3.2]. We will state this result in larger generality in Chapter 3.

Denition 1.3.2.3. Fix two bounded graded norms ζ • , ζ • ∈ N • (V • ).
We dene:

• for p ∈ [1, ∞), their asymptotic d p -distance by:

d 1 (ζ • , ζ • ) = ˆR |λ| dσ(ζ • , ζ • ); • their asymptotic d ∞ -distance by: d ∞ (ζ • , ζ • ) = sup k∈N * k -1 d ∞ (ζ k , ζ k );
• their asymptotic relative volume by:

vol(ζ • , ζ • ) = ˆR λ dσ(ζ • , ζ • ).
One then sees that the asymptotic d p -distances may be recovered as the limit of the nite-dimensional distances

k -1 d p (ζ k , ζ k ) p = k -1 ˆR |λ| p dσ(ζ k , ζ k ).
Similarly, the asymptotic relative volume is recovered as

vol(ζ • , ζ • ) = lim k k -1 vol(ζ k , ζ k ).
It therefore has the same algebraic properties of nite-dimensional volumes, i.e. the cocycle property and antisymmetry.

Note that the d p "distances" above are merely pseudodistances: for example, since for any two norms

ζ k , ζ k on a xed V k , d 1 (ζ k , ζ k ) ≤ d ∞ (ζ k , ζ k ),
then if two bounded graded norms have at most subexponential growth in k,

we have

d 1 (ζ • , ζ • ) ≤ lim k k -1 d ∞ (ζ k , ζ k ) = 0.
Even worse: there exist bounded graded norms such that k

-1 d ∞ (ζ k , ζ k ) → C > 0 but d 1 (ζ • , ζ • ) = 0,
ζ • ∼ ζ • , if and only if d 1 (ζ • , ζ • ) = 0.
We then have that

(N • (L)/ ∼, d 1 )
with the induced d 1 distance, is a bona de metric space.

Remark 1.3.2.5. One has that ζ • ∼ ζ • if one of the three following equivalent conditions is realized:

• for some p ∈ [1, ∞), ´R |λ| p dσ(ζ • , ζ • ) = 0; • for all p ∈ [1, ∞), ´R |λ| p dσ(ζ • , ζ • ) = 0; • the asymptotic spectral measure σ(ζ • , ζ • ) is the Dirac measure δ 0 .
This is proven in the trivially valued case in [BJ18a, Section 3.6], and the proofs are identical in the nontrivially valued case. In particular, this justies our choice of notation for ∼, which does not emphasize the choice of a p ∈ [1, ∞).

Remark 1.3.2.6. Note that, since d ∞ is dened as a sup rather than as a limit, it denes a genuine distance on N • (L).

Finally, we check that bounded graded norms remain bounded graded after piecewise ground eld extension. This is a result of Boucksom-Eriksson:

Lemma 1.3.2.7 ([BE, L9.4]). Set ζ • , ζ • ∈ N • (V • ). Let L/K be a complete
eld extension, and consider the sequences of ground eld extensions

ζ L,• , ζ L,• .
Then, those sequences are bounded graded norms, and furthermore

σ(ζ L,• , ζ L,• ) = σ(ζ • , ζ • ),
which implies stability of the asymptotic volumes and d p -distances under base change.

Geodesics between bounded graded norms.

In [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], we extend the classical results of Section 1.2.5 to the context of bounded graded norms. Namely, given two bounded graded norms ζ 0

• and ζ 1 • , one wonders whether there exists a geodesic of graded norms in N • (V • ) for the asymptotic d 1 -distance (or d p , p < ∞). It seems obvious to consider, for all k, the norm geodesic ζ t k joining ζ 0 k and ζ 1 k in V k , and to set, for all t,

ζ t • = (ζ t k ) k .
There are two points to show here: submultiplicativity, and geodesicity. The latter is rather simple: by denition of our norm geodesics, we have that, for

t, t ∈ [0, 1], d p (ζ t k , ζ t k ) = |t -t | • d p (ζ 0 k , ζ 1 k )
, so that geodesicity follows upon taking the limit in k. Showing submultiplicativity is a bit trickier.

Theorem 1.3.3.1. For all t ∈ [0, 1], the sequence of norms ζ t

• dened above is submultiplicative. Proof of Theorem 1.3.3.1. We assume at rst that all norms involved are diagonalizable. We start with the following case: let v m belong to a basis of V m orthogonal for ζ 0 m and ζ 1 m . Dene in in the same way s n ∈ V n . We then have that

ζ t m+n (v m • v n ) ≤ ζ 0 m+n (v m • v n ) 1-t ζ 1 m+n (v m • v n ) t ≤ ζ 0 m (v m ) 1-t ζ 0 n (v n ) 1-t ζ 1 m (v m ) t ζ 1 n (v n ) t = ζ t m (v m ) • ζ t n (v n ),
where we have used log-convexity (Lemma 1.2.5.3) in the rst inequality, submultiplicativity of the endpoints in the second inequality, and nally the fact that v m and v n belong to bases codiagonalizing the endpoints, so that

ζ t m (v m ) = ζ 0 m (v m ) 1-t ζ 1 m (v m ) t
, and the same holds for n.

To pass to the general case, write

v m = a i v m,i , v n = b j v n,j ,
in their adapted bases, and note that

ζ t m+n (v m • v n ) ≤ max i,j |a i | • |b j | • ζ t m+n (v m,i • v n,j ) ≤ max i,j |a i | • |b j | • ζ t m (v m,i ) • ζ t n (v n,j ) ≤ (max i |a i | • ζ t m (v m,i )) • (max j |b j | • ζ t n (v n,j )) ≤ ζ t m (v m ) • ζ t n (v n ).
The second inequality follows from the result we just proved, which applies to the v m,i and the v n,j ; the fourth inequality follows from the fact that those bases diagonalize ζ t m and ζ t n . This proves the desired result.

Finally, if the norms are not diagonalizable, pick, for i = 0, 1, ε > 0, and all m ∈ N * , diagonalizable norms

ζ i,ε m such that d ∞ (ζ i,ε m , ζ i m ) < ε.
Those always exist by d ∞ -density of the set of diagonalizable norms on a nite-dimensional K-vector space. By the distortion characterization of d ∞ , we then have that, for all m:

e -ε ζ i m ≤ ζ i,ε m ≤ e ε ζ i m .
(1.1)

Pick sections v m ∈ V m , v ∈ V . We then have that ζ i,ε m+ (v m • v ) ≤ e ε ζ i m+ (v m • v ) ≤ e ε ζ i m (v m ) • ζ i (v ) ≤ e 3ε ζ i,ε m (v m ) • ζ i,ε (v ).
We have used the right-hand side of (1.1) for the rst inequality; submultiplicativity of ζ i

• for the second inequality, and nally the left-hand side of (1.1) for the third one. Multiplying both sides by e 3ε we then have that

e 3ε ζ i,ε m+ (v m • v ) ≤ e 6ε ζ i,ε m (v m ) • ζ i,ε (v ),
i.e. the sequence of norms e 3ε ζ i,ε

• is submultiplicative. As d ∞ (e 3ε ζ i,ε m , ζ i m ) = 3ε + d ∞ (ζ i,ε m , ζ i m ) < 4ε,
one can see the e 3ε ζ t,ε m to be the norm geodesics joining the e 3ε ζ i,ε m , and also actually e 3ε times the geodesic joining the ζ i,ε m ). We nd

d ∞ (e 3ε ζ t,ε m , ζ t m ) ≤ (1 -t)d ∞ (e 3ε ζ 0,ε m , ζ 0 m ) + td ∞ (e 3ε ζ 1,ε m , ζ 1 m ) < 4ε,
thanks to metric convexity of d ∞ (Lemma 1.2.5.10). This states that e 3ε ζ t,ε m converges pointwise to ζ t m for all m. As e 3ε ζ i,ε m is diagonalizable for all m, by the previous case, e 3ε ζ t,ε

• is submultiplicative, and in particular we have

e 3ε ζ t,ε m+ (v m • v ) ≤ e 6ε ζ t,ε m (v m )ζ t,ε (v )
Using the pointwise convergence found above to pass to the limit as ε → 0, this proves the Theorem.

We remark that the last part of the above proof also shows the following:

Proposition 1.3.3.2. Let ζ • be a bounded graded norm on V • . Then, there exist bounded graded norms ζ ε • on V • , for all ε > 0, satisfying the following properties:

• ζ ε m is diagonalizable for all m; • d ∞ (ζ ε m , ζ m ) < ε for all m. 1.3.4 Completeness of N • with respect to the d ∞ dis- tance.
To conclude this Chapter, we prove the following unpublished result.

Theorem 1.3.4.1. The metric space

(N • (V • ), d ∞ ) is complete. Proof. Let k → ζ k
• be a Cauchy sequence of bounded graded norms with respect to the d ∞ distance. Then, for all ε > 0, there exist an k ε > 0 such that for all m, n > k ε ,

sup p∈N * k -1 d ∞ (ζ m p , ζ n p ) < ε, which implies that for all integers k, d ∞ (ζ m p , ζ n p ) < p • ε.
In other words, the sequence k → ζ k p is Cauchy with respect to the d ∞ distance on N (V p ). Since spaces of norms on nite-dimensional vector spaces are complete with respect to d ∞ , there exists a d ∞ -limit ζ p ∈ N (V p ) to this sequence. The family of norms ζ

• : p → ζ p ∈ N (V p ) is then by denition a d ∞ -limit to the sequence of bounded graded norms (ζ k • ) k .
We claim that ζ • is bounded graded. Boundedness follows from the triangle inequality, and we must therefore show that ζ • is graded. This will follow from the "submultiplicativity trick" from the proof of submultiplicativity of geodesics in N • (L). We thus rst choose an ε > 0 and pick k large enough so that

d ∞ (ζ k • , ζ • ) < ε.
For all integers p, this yields

d ∞ (ζ k p , ζ p ) < p • ε,
and using the fact that d ∞ is the maximal exponential distorsion between two norms, we nd

e -p•ε ζ k p ≤ ζ k ≤ e p•ε ζ k p ,
for all p uniformly. We now pick p, q integers, and elements v p ∈ V p , v q ∈ V q . Using submultiplicativity of ζ k • , we nd:

ζ p+q (v p • v q ) ≤ e p•ε ζ k p+q (v p • v q ) ≤ e p•ε ζ k p (v p )ζ k q (v q ) ≤ e 3p•ε ζ p (v p )ζ q (v q ).
Since, for all ε > 0, we can nd a k such that this works, this proves that

ζ p+q (v p • v q ) ≤ ζ p (v p )ζ q (v q ).
Remark 1.3.4.2. It is expected that the quotient spaces N • (V • )/ ∼ are not complete with respect to any of the d p distances, p < ∞. In the case where K is trivially valued and p = 1, the completion has been characterized by ).

Chapter 2 Geometric preliminaries.

Conventions.

Throughout this Chapter, K will be a complete valued eld. A variety X over K will be a geometrically integral, separated scheme, of nite type over K.

2.1 Analytic geometry over non-Archimedean elds.

2.1.1 The Berkovich analytication.

In analogy with the fundamental GAGA principle of Serre ([SerGAGA]), Berkovich denes in [BerkBook] an analytication functor sending a Kvariety X to its Berkovich analytication X an . It is a K-analytic space in the sense of [BerkBook], but we shall not develop this aspect here, and rather focus on the topological space X an .

The ane case. Assume X to be the spectrum Spec A of an algebra A of nite type over K. Then, the underlying set of (Spec A) an is the set of multiplicative seminorms on A which extend the absolute value on K.

The topology on (Spec A) an is the topology of pointwise convergence, that is: the coarsest topology such that, for all a ∈ A, the evaluation map

X an | • | A → |a| A is continuous.
This construction gives a natural kernel map from (Spec A) an to Spec A, The general case. If X is an abstract variety, we proceed by gluing the ane construction above. We cover X by anes U i = Spec A i . By [Berk93, Proposition 1.3.3], one can glue together the U an i provided each U ij := U i ∩U j is ane. This is ensured by separatedness of X. The kernel maps ker i : U an i → U i glue together as well to a global kernel map ker : X an → X.

We then have the following:

Theorem 2.1.1.1 ([BerkBook, Theorems 3.4.1, 3.4.8, 3.5.1, 3.5.3]). Assume X to be a connected projective K-variety. The mapping X → X an realizes an equivalence of categories between the categories of coherent sheaves on X and X an . Furthermore, X an is compact, connected, and Hausdor.

Example 2.1.1.2. Assume K to be C endowed with its Archimedean absolute value. Then, X an is isomorphic to the usual analytication of the complex variety X in the sense of Serre ( [SerGAGA]).

Example 2.1.1.3. Assume K to be trivially valued. Then, the points of X an are identied with the set of semivaluations on X, i.e. the set of valuations on K(Y ) for all irreducible subvarieties Y of X.

Example 2.1.1.4. If K is non-Archimedean, in general, one identies X an with pairs (a, ν a ), where a is a scheme point of X, and ν is a valuation on the residue eld of that point, extending the valuation -log | • | on K.

Example 2.1.1.5. If X = Spec K, then its analytication is again a point, corresponding to the absolute value on K.

This analytication is, as mentioned, functorial: this implies that, given a morphism of K-schemes f : X → Y , one obtains a morphism f an : X an → Y an compatible with the analytic structure morphisms.

2.1.2 Models and divisorial points.

We now assume K to be non-Archimedean, and X to be a connected projective K-variety.

Denition 2.1.2.1. A model of X is the data of:

1. a at scheme X of nite type over the valuation ring K • ;

2. an isomorphism X × K • K → X as schemes over K.

Example 2.1.2.2. If K is trivially valued, the only model of X is X itself, up to automorphisms. We will need an alternative denition, that of a test conguration. This will be considered in the next Subsection.

For the remainder of this Subsection, we assume K to be nontrivially valued.

Denition 2.1.2.3. As a scheme, Spec K • has two points: the generic point corresponding to the ideal {0}, with residue eld K, and a closed point, the special point, corresponding to K •• , with residue eld K. Changing the base to K, resp. K, amounts to taking the generic bre X K , resp. special bre or central bre X s (we will sometimes also denote it by X 0 ). Remark 2.1.2.4. Note that, if K is discretely valued, K • is a discrete valuation ring. On the other hand, if K is densely valued, K • can never be Noetherian. This implies that, over densely valued elds, we will have to work with non-Noetherian schemes.

Denition 2.1.2.5. We say that a model X dominates (resp. properly dominates) a model X if there exists a morphism (resp. a proper morphism) X → X . Any two models may always be jointly dominated by a third.

Denition 2.1.2.6. Let X be a model of X. Let x = (a, ν a ) be a point of X an , identied with a scheme point of X together with a valuation on the residue eld at a extending the underlying absolute value on K. Let R a denote the associated valuation ring in the residue eld K(ker a) at ker a.

The valuative criterion for properness ensures that there exists a unique lift of the map Spec R a → Spec K • to Spec R a → X . We denote by red X (a) the image of the closed point K •• by this lift. The map red X so dened is called the reduction map associated to X . We shall say that a point in x ∈ X an is divisorial or Shilov, and we will write x ∈ X div , if there exists a model X such that red X (x) is a generic point of the special bre X s . The set of divisorial points is dense in X an .

Example 2.1.2.7. If K is discretely valued, and if we pick a normal model X of X, then X s decomposes as a Weil divisor

X s = a i E i ,
each E i an irreducible component of the special bre; furthermore, the E i each dene a unique divisorial point x E i , such that red X (x E i ) the generic point of E i , dened as follows. Let r be the smallest nonzero positive element in the value group of K. Then,

x E i = r ord E i (•)/a i ,
where a i is the coecient of E i in the Weil decomposition of the special bre.

We now assume K to be trivially valued. We then dene the set X div of divisorial points inside the analytication X an of X with respect to the trivial absolute value on K, to be the set of points of the form

c • ord E (•),
where c is a positive rational constant, and E is a prime divisor in a projective,

normal birational model E ⊂ Y → X of X.
It is again dense in X an , provided X has nonzero dimension. We will not study here the notion of test conguration, which plays the role of models in the trivially-valued case.

2.2 Metrization of analytications of line bundles.

2.2.1 Line bundles and models thereof.

We will x X as usual, and let L be a line bundle on X. We will denote by

H 0 (X, L),
or simply H 0 (L), its space of sections. Its dimension will be written as h 0 (X, L).

Finally, we use additive notation for tensor powers of line bundles, which means that we write

L ⊗k ⊗ M ⊗-1 = kL -M,
given another line bundle M on X.

We shall occasionally speak of Q-line bundles, by which we mean formally an element of Pic(X) ⊗ Q. For all divisible enough m, mL is therefore a genuine line bundle on X.

We will say that a line bundle L is:

1. very ample if its sections dene a closed immersion of X into P h 0 (X,L)-1 ;

2. ample if mL is very ample for large enough m;

3. basepoint-free if there is no point x ∈ X such that all sections of L vanish at x; 4. semiample if mL is basepoint-free for large enough m.

We will say that a Q-line bundle L is:

1. ample if mL is very ample for divisible enough m;

2. semiample if mL is basepoint-free for divisible enough m.

We now turn to polarized models.

Denition 2.2.1.1. Let L be a line bundle on X. A model (X , L) of (X, L)

is the data of:

1. a model X of X, projective over K • ; 2. a line bundle L on X ;

3. an isomorphism L K L compatible with the isomorphism X K X.

One then says that L is a model of L determined on X . Denote by π :

X → Spec K • the structure morphism. We will say that a model (X , L) of (X, L) is:

2. ample if mL is π-very ample for large enough m;

3. basepoint-free if the morphism π * π * L → L is surjective;

4. semiample if mL is basepoint-free for large enough m;

5. nef if L • C ≥ 0 for all projective curves in the special bre X s . Denition 2.2.1.2. Let L be a Q-line bundle on X. A Q-model (X , L) is the data of: 1. a model X of X, projective over K • ;
2. a Q-line bundle L on X such that (X , mL) is a model of (X, mL) for divisible enough m.

One then says that L is a Q-model of L determined on X . We will say that a Q-model (X , L) of (X, L) is ample (resp. semiample) if for divisible enough m, mL is ample (resp. semiample).

Example 2.2.1.3. If L = O X , one can identify models of L determined on X with vertical Cartier divisors on X , i.e. Cartier divisors supported in the special bre of X .

2.2.2 Metrics over analytications of K-line bundles.

For x ∈ X an , we will denote by H(x) the completion of the residue eld at x, endowed with its canonical absolute value.

|s U | φ : U an (s U ) an ---→ (L| U ) an |•| φ -→ R ≥0
is continuous. We will denote the space of continuous metrics on L an by C 0 (L an ). More generally, we can dene a singular metric to be a sum of the form φ + u, where φ is a continuous metric on L an , and u is any function u : X an → [-∞, ∞). Whenever we shall speak of metrics in the remainder of this thesis, we will assume that they can be singular. Similarly, we may dene:

1. the spaces L ∞ (L an ) and L ∞ loc (L an ) of bounded, resp. locally bounded metrics on L an , which are written φ + u for φ ∈ C 0 (L an ) and u a function X an → R which is bounded, resp. locally bounded; 2. given a Radon measure µ with nite total mass on X an , the spaces L 1 (µ, L an ) and L 1 loc (µ, L an ) of integrable, resp. locally integrable metrics on L an with respect to µ, which are written φ + u for φ ∈ C 0 (L an ) and u a function X an → R in L 1 (µ, X an ), resp. L 1 loc (µ, X an ).

Fix a reference continuous metric φ ref on L an . We also use additive conventions for metrics, which implies that, given two line bundles L and M on X,

• given metrics φ on L an and φ on M an , the induced metric on (kL-M ) an is written as kφ -φ ;

• we may identify a metric φ with the (possibly singular) function -log |1| φ-φ ref on X an , since a metric on (O X ) an canonically identies with a function on X an .

In particular, by noticing that two metrics φ, φ on L an transform as

| • | φ = | • | ψ e φ -φ
, we can see the spaces C 0 (L an ), L ∞ (L an ), L ∞ loc (L an ), L 1 (µ, L an ), and L 1 loc (µ, L an ) as ane spaces modelled on respectively C 0 (X an ), L ∞ (X an ), L ∞ loc (X an ), L 1 (µ, X an ), and L 1 loc (µ, X an ).

We shall write for the rest of this Chapter |s| = |s| φ ref for any section s ∈ H 0 (L).

Model metrics.

Given an ample line bundle L on X, we now dene a class of metrics dened using very explicit algebraic data: that given by a model (X , L) of (X, L)generalizing the idea that a K • -lattice in a K-vector space canonically denes a norm. Throughout this Subsection, we will use many results from (and follow) [BE, Section 5.3].

Denition 2.2.3.1. Let (X , L) be a model of (X, L). We dene the model metric φ L ∈ C 0 (L an ) as follows. Pick x ∈ X an , and recall that there is associated to X its reduction map red X : X an → X s . Pick a section s U of L over a Zariski neighbourhood of red X (x), which does not vanish at the reduction of x. Using the identication X K X, we may then consider the analytication (X ∩ U ) an x, and notice the analytication (s U ) an to be nonvanishing at x. We may therefore require that |s U | φ L ≡ 1 on (X ∩ U ) an . This gives a well-dened continuous metric on L an : given another section s U of L over U nonvanishing at red X (x), then there exists a unit

u ∈ O × X (U ) such that s U = u•s U , which implies that |s U | φ L = |s U | φ L ≡ 1.
While this denition seems rather abstract, we will see in the next Section that, if L has nice positivity properties, then model metrics on L an can be recovered as more familiar objects.

Example 2.2.3.2. If K is trivially valued, and (X, L) has no automorphisms, so that the only model of (X, L) is itself, the associated model metric is the trivial metric on L an . This gives some insight into how to generalize certain results from the trivially valued to the nontrivially valued case: model metrics can be seen as playing the role of the trivial metric in the case there is no canonical trivial model.

In order to dene model metrics associated to Q-models, we shall need the following result:

Lemma 2.2.3.3 ([BE, Lemma 5.10]). If (X , L) is a model of (X, L), then:

1. for all integers m, φ mL = mφ L ;

2. if a model (Y, M) dominates a model (X , L) via π : Y → X , then φ π * L = φ L ;
3. in particular, if (X , L) and (X , L ) are two models of (X, L), then φ L = φ L if and only if one can dominate both models by a third model (Y, M) with π : Y → X , π : Y → X , and such that π * L = π * L .

Denition 2.2.3.4. Let L be a Q-line bundle, and (X , L) be a Q-model of (X, L). We dene φ L = m -1 φ mL for any m divisible so that mL is a line bundle; this is well-dened by the rst point of the above Lemma.

Given L, M two Q-line bundles on X, and model metrics φ L , φ M on L an and M an , as well as a rational r ∈ Q, then rφ L -φ M is a model metric on rL -M .

Remark 2.2.3.5. We may now begin to explain the sibylline remark at the beginning of our talk on models. Given a model (X , L) of (X, L), the space of sections H 0 (X , L) determines a K • -lattice the K-vector space H 0 (X, L), and therefore a lattice norm on N (H 0 (X, L)). We will see shortly how to associate to such norms a metric on L an , and we will in fact see that it will coincide with the metric φ L in nice cases.

Metrization of the canonical line bundle, after Temkin.

A surprising fact of life in the non-Archimedean world is that the (powers of the) canonical line bundle can always be endowed with a special metric, generalizing the log-discrepancy on the valuation space of a variety. This construction is due to Temkin ( [Tem]). The details in the general case are rather complicated and out of the scope of this manuscript, and we therefore direct the reader to Temkin's original article (see also an exposition in Stevenson's thesis [StevThesis]). We will therefore focus on the special cases of a trivially-valued eld, and a discretely-valued eld of equal characteristic zero. Other references for this Subsection are [START_REF] Jonsson | Valuations and asymptotic invariants for sequences of ideals[END_REF], [START_REF] Boucksom | Tropical and non-Archimedean limits of degenerating families of volume forms[END_REF], [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF].

The trivially-valued case. If the base eld is trivially-valued, any line bundle L on a projective K-variety X admits a trivial metric φ triv , characterized as the model metric associated to the unique (trivial) model of (X, L), by which one may canonically identify metrics on L an with functions on X an .

Temkin's metric φ Tem on K an X is then characterized as

φ Tem = φ triv + A X ,
where φ triv is thus the trivial metric on K an X , and A X is the log-discrepancy function on X an , whose denition we recall now. Given a divisorial point x on X an , which is by denition an exponential of a valuation of the form

c • ord E (•),
where c ∈ Q >0 , and E is a prime divisor in a projective, normal birational model π : Y → X. Then, we set

A X (x) = c • (1 + ord E (K Y -π * K X )).
This is extended to a lsc function on all of X an , which is characterized as the largest lsc extension of A X , as in [BJ18a, Theorem 2.1]. As we shall now see, a similar characterization holds in the discretely-valued case.

The discretely-valued, equal characteristic zero case. There are now no longer any "canonical" models of the canonical bundle, and therefore now longer a trivial metric. Pick therefore a proper model X of X, and denote by s : X → Spec K • its structure morphism.

Ingredient 1: the model metric. As in [MN12, (4.1.1)] and [BJ17, (5.

3)],

one can dene the relative canonical divisor K X / Spec K • associated to the morphism s, and thus the relative log canonical divisor

K log X / Spec K • = K X / Spec K • + (X s ) red -X s ,
where (X s ) red denotes the reduction (in the sense of scheme theory) of the special bre of X . Those are Weil divisors; if we furthermore assume one (hence both) to be (Q-)Cartier, the relative log canonical divisor is then a

model of K X , dening a model metric φ K log X / Spec K • .
Ingredient 2: the log-discrepancy function. To X is also associated a logdiscrepancy function A X , characterized on divisorial points as follows: let ρ : Y → X be some model dominating X . Since any two models can be jointly dominated by a third model, any point in cX div is a valuation ν E associated to a divisor a the central bre Y s = i a i E i of such models. The value of the log discrepancy function against such a divisorial point is thus fully characterized via the formula

K Y + Y s = ρ * (K X + X s ) + i A X (ν E i )a i E i .
This is then extended as before to a maximal lsc function A X on X an , as in [BJ17, (5.6)]. Temkin's metric is recovered as

φ Tem = φ K log X / Spec K • + A X ,
A striking observation is that this is independent of the choice of a model X (using the formula above characterizing A X , and the fact that the model metric of a dominating model is the model metric of the original model), thus giving a very natural new metric on K an X .

Pluripotential theory over nontrivially valued elds

2.3.1 Fubini-Study and plurisubharmonic metrics.

In the complex setting, due to the work of Demailly, the class of psh metrics on a holomorphic line bundle L is characterized as the smallest class stable under nite maxima, addition of constants, and decreasing limits, of metrics containing all Fubini-Study metrics, that is, metrics of the form φ = log 1 2k

h 0 (kL) j=1 |s j | 2 e 2λ j ,
where the s j are a basepoint-free basis of sections of kL, and the λ j are real constants. We will therefore dene non-Archimedean metrics similarly, using an adequate "non-Archimedean version" of Fubini-Study metrics.

Denition 2.3.1.1. Let L be a Q-line bundle on X. We dene a Fubini-Study metric on L an to be a metric of the form φ = 1 k log max j=1,...,h 0 (kL)

|s j |e λ j ,
where the s j are a basepoint-free basis of sections of kL, and the λ j are real constants. We furthermore say that such a metric is:

• a Q-Fubini-Study metric if the λ j are all rational;

• a K-rational Fubini-Study metric if the λ j belong to the value group of K;

• a pure Fubini-Study metric if the λ j are all equal to zero.

We shall write FS(L an ), FS Q (L an ), FS Γ(K) (L an ), and FS 0 (L an ) for those classes, respectively. More generally, as in [BE], we can dene the class FS Γ (L an ) of Γ-Fubini-Study metrics, for any subgroup Γ of the real line.

We look at some immediate properties of Fubini-Study metrics.

Proposition 2.3.1.2 ([BE, Proposition 5.4]). Given a subgroup Γ of R, two line bundles L and M on X, and a positive integer k, then:

1. FS Γ (L an ) + FS Γ (M an ) ⊂ FS Γ (L an + M an ); 2. FS Γ (mL an ) = mFS Γ (L an ); 3. FS Γ (L an ) = FS Q(Γ+Γ(K)) (L an ); 4. FS Γ (L an ) is stable under nite maxima.

Remark 2.3.1.3. As a consequence of the third point above, we have

FS 0 (L an ) = FS Q(Γ(K)) (L an ).
This implies that, after a large enough ground eld extension, any Fubini-Study metric becomes pure.

As it turns out, "positive" model metrics are in correspondence with Fubini-Study metrics.

Theorem 2.3.1.4 ([BE, Theorem 5.14, Corollary 7.9]). Let L be a Q-line bundle. Set a metric φ ∈ C 0 (L an ). We then have that:

• φ is a pure Fubini-Study metric if and only if it coincides with the model metric φ L associated to a semiample Q-model L of L; • φ is a Fubini-Study metric if and only if it coincides with the model metric φ L associated to a nef Q-model L of L.

We now take the statement of the complex regularization Theorem to be our denition of a non-Archimedean plurisubharmonic metric:

Denition 2.3.1.5. A metric φ on the analytication L an a Q-line bundle L is plurisubharmonic or psh if it can be written as a decreasing net of Fubini-Study metrics on L, and is not identically -∞. We denote by PSH(L an ) the class of plurisubharmonic metrics on L an .

Proposition 2.3.1.6 ([BJ21, Theorem 4.5]). Let L, M be Q-line bundles.

The class PSH satises the following properties: 1. it contains all Fubini-Study metrics on L an ; 2. it is stable under: (a) taking nite maxima;

(b) addition of a real constant; (c) limits of decreasing nets; 3. the convex combination of two psh metrics on L an is a psh metric on L an ; 4. PSH(L an ) + PSH(M an ) ⊂ PSH(L an + M an ); 5. if a net of psh metrics on L an converges uniformly to a limit metric, then this limit metric is psh.

Note that most of those properties follow directly from the denition, but 2(c) is not immediate, as we consider not only decreasing sequences, but decreasing limits of decreasing nets. We have cited [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], which details the trivially valued case, but the proofs follow the same lines in the nontrivially valued case.

As desired, we then have a similar characterization as in the complex case:

Proposition 2.3.1.7 ([BJ21, Corollary 4.16]). The class PSH(L an ) is the smallest class of metrics on L satisfying properties 1. and 2. above.

We endow PSH(L an ) with the topology of pointwise convergence on the set of divisorial points X div ⊂ X an : a net φ i in PSH(L an ) is said to converge to φ ∈ PSH(L an ) if and only if φ i (x) → φ(x) for all x ∈ X div . It is important to note that psh metrics are uniquely determined by their restriction to X div .

The Fubini-Study and supnorm operators.

We now introduce the Fubini-Study operators, which give a way to turn norms into Fubini-Study metrics:

Denition 2.3.2.1. We dene the (m-th) Fubini-Study operator as follows: There is a nice way to compute the Fubini-Study operators restricted to each N diag (H 0 (mL)). Indeed, assume (s i ) diagonalizes a norm ζ on H 0 (mL) for some large m. We then have that:

FS m : N (H 0 (mL)) → C 0 (L an ), ζ → FS m (ζ) = 1 m log sup s∈H 0 (mL)-{0} |s| ζ(s) .
FS m (ζ) = 1 m log max i |s i | ζ(s i ) .
This remark combined with the above Lemma allows us to ensure that our Fubini-Study operators work as intended:

Corollary 2.3.2.3. A metric in the image of some Fubini-Study operator is Fubini-Study. Proof. This is only nontrivial if K is not maximally complete, since not all norms on K-vector spaces are diagonalizable. But we can pick a maximally complete extension L of K, where all norms on K-vector spaces are diagonalizable, and the above Lemma proves our result.

Remark 2.3.2.4. It follows that pure Fubini-Study metrics are characterized as those metrics φ which belong to the image of some Fubini-Study operator restricted to the set of lattice norms on some space of plurisections of L.

We now construct operators going the other way around: from the space of continuous metrics on L an to spaces of norms on the H 0 (mL).

Denition 2.3.2.5. The m-th supnorm operator N m sends φ ∈ C 0 (L an )

to the norm on H 0 (mL) dened as

N m (φ)(s) = sup X an |s| mφ ,
for s ∈ H 0 (mL).

It it straightforward from the denitions to see the following:

Proposition 2.3.2.6 (Lipschitz-like properties of FS and N). Consider two continuous metrics φ, φ on L, and two norms ζ and ζ on some H 0 (mL).

We then have that:

• d ∞ (N m (φ), N m (φ )) ≤ m • sup X an |φ -φ|; • sup X |FS m (ζ ) -FS m (ζ)| ≤ m -1 • d ∞ (ζ, ζ ).

Plurisubharmonic envelopes.

Given a function f : R d → R, one can construct its convex envelope as the largest convex function bounded above by f . Given a metric on a holomorphic line bundle, one can again construct its plurisubharmonic envelope; it is a classical result from pluripotential theory that this envelope is continuous provided the original metric is continuous. We look here into similar problems in the non-Archimedean world. Throughout this Subsection, L is a line bundle on a projective K-variety X, with K non-Archimedean.

Denition 2.3.3.1. Let φ be a bounded metric on L an . The psh envelope of φ is dened as

P (φ) = sup{φ ∈ PSH(L an ), φ ≤ φ}.
The regular psh envelope of φ is dened as

Q(φ) = sup{φ ∈ PSH(L an ) ∩ C 0 (L an ), φ ≤ φ}.
Denition 2.3.3.2 (Continuity of envelopes). We say that the pair (X, L) admits continuity of envelopes if the following property holds true:

• if φ is a continuous metric on L, then P (φ) is continuous.

Example 2.3.3.3. By [START_REF] Boucksom | Singular semipositive metrics on line bundles on varieties over trivially valued elds[END_REF], a smooth, projective variety X dened over any eld K which satises all of the following properties:

• K is of equal characteristic 0;

• K is either trivially or discretely valued, admits continuity of envelopes for any ample line bundle L over X. Furthermore, by [START_REF] Gubler | Continuity of plurisubharmonic envelopes in non-Archimedean geometry and test ideals[END_REF], continuity of envelopes also holds:

• for any line bundle on a curve, over any eld (from the work of Thuillier);

• for all line bundles on a d-dimensional variety X over K, where K is a discretely valued eld of positive characteristic p, provided we have resolution of singularities over K in dimension d + 1.

It is expected in general that, for X unibranch (in particular, normal) and L ample, over any non-Archimedean eld, continuity of envelopes holds for (X, L).

One can show the following:

Theorem 2.3.3.4 ([BE, Theorem 7.26]). Let φ be a bounded on L an . Then,

lim m FS m (N m (φ)) = Q(φ).
Here, the limit is taken over a divisible sequence of integers so that the associated power of L is always globally generated. Furthermore, the convergence is uniform if and only if Q(φ) is continuous.

We then have that:

Proposition 2.3.3.5. The following are equivalent: 1. continuity of envelopes holds for (X, L); 2. for all continuous metrics φ on L an , P (φ) = Q(φ); 3. for all continuous metrics φ on L an , lim m FS m (N m (φ)) = P (φ); 4. the sequence m → FS m (N m (φ)) converges uniformly; 5. given any family (φ i ) i∈I of psh metrics on L an uniformly bounded above, the usc regularization of the upper envelope usc(sup

i∈I φ i )
is psh. Proof. That (2) and (3) are equivalent follows from Theorem 2.3.3.4. That

(1) and (2) are equivalent is immediate. That (1) and (4) are equivalent follows from Theorem 2.3.3.4 again and Dini's Lemma. Finally, equivalence of (1) and (5) is the statement of [BE,Lemma 7.29]. This is the foundation for the quantization principle: approximating bounded psh metrics from below with metrics coming from norms (our space of Fubini-Study metrics!), which has proven its usefulness multiple times in complex geometry.

Denition 2.3.3.6. The asymptotic Fubini-Study operator is dened on the set of bounded graded norms on L as the usc regularization

FS(ζ • ) = usc lim m FS m (ζ m ) .
Remark 2.3.3.7. The asymptotic Fubini-Study operator is well-dened and denes a bounded psh metric provided that (X, L) admits continuity of envelopes, as, by Fekete's lemma,

lim m FS m (ζ m ) = sup m FS m (ζ m ).
We shall also dene its brother: Denition 2.3.3.9. We say that a plurisubharmonic metric φ on L an is regularizable from below, and we write φ ∈ PSH ↑ (L an ), if and only if φ is the pointwise limit on X div of an increasing net of Fubini-Study metrics, equivalently of an increasing net of continuous, psh metrics.

Remark 2.3.3.10. We then have that φ is the usc regularized supremum of such a net. Furthermore, φ ∈ PSH ↑ (L an ) if and only if φ = usc Q(φ).

We then show that PSH ↑ (L an ) coincides with the image of the asymptotic Fubini-Study operator.

Theorem 2.3.3.11. A metric φ belongs to PSH ↑ (L an ) if and only if there exists a bounded graded norm

ζ • such that FS(ζ • ) = φ.
Proof. Assume φ is the image of some bounded graded norm ζ • by the asymptotic Fubini-Study operator, i.e. φ = usc (lim m FS m (ζ m )). In particular, φ is psh, by continuity of envelopes. By the remark above, it is then enough to show that φ = usc Q(φ), which is clear by construction.

We now assume φ to be regularizable from below. By Theorem 2.3.3.4,

Q(φ) = lim m FS m (N m (φ)). Then, by denition, usc Q(φ) = FS(N • (φ)). Since φ ∈ PSH ↑ (L an ), we have that φ = usc Q(φ), thus φ = FS(N • (φ)),
which proves the Theorem.

The purpose of Chapter 3, and indeed the paper [START_REF] Reboulet | The asymptotic Fubini-Study operator over general non-Archimedean elds[END_REF], is to reverse this characterization: we show that the asymptotic Fubini-Study operator in fact denes a bijection between PSH ↑ (L an ), and the space of bounded graded norms on L modulo asymptotic equivalence.

2.4

The space of nite-energy plurisubharmonic metrics.

We conclude this preliminary Chapter by studying the space of non-Archimedean nite-energy metrics, in analogy with the works of Darvas and Berman-Boucksom-Eyssidieux-Guedj-Zeriahi in the complex case ([Dar17], [BBEGZ], [BBGZ]). We will again focus on the case where the base eld K is nontrivially valued; the trivially-valued situation has been treated in much detail in [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF]. We again assume, throughout this Section, that X is a projective K-variety.

Monge-Ampère operators and Deligne pairings.

We begin with a discussion of Monge-Ampère operators in non-Archimedean pluripotential theory. Using either intersection pairings ( [Gub], [BE]), or the theory of dierential forms on Berkovich spaces developed by A. Chambert-Loir and A. Ducros in [START_REF] Chambert | Formes diérentielles réelles et courants sur les espaces de Berkovich[END_REF]5,6], one may dene a Radon probability measure associated to d = dim X continuous plurisubharmonic metrics φ 1 , . . . , φ d on analytications of ample line bundles L 1 , . . . , L d over the analytic space X an , denoted

MA(φ 1 , . . . , φ d ) = V -1 • dd c φ 1 ∧ • • • ∧ dd c φ d ∧ δ X ,
with V the intersection number of the L i . For short, if (e.g.) the metric φ 1 appears n times in the expression, we write MA(φ

(n) 1 , . . . ) = V -1 • (dd c φ 1 ) n ∧ • • • ∧ δ X ,
and so on; and we set

MA(φ) = MA(φ (d) ) = V -1 • (dd c φ) d ∧ δ X .
Proposition 2.4.1.1 ([BE, P8.3(iv)]). Let K /K be a non-Archimedean eld extension. Consider the cartesian diagram:

X an K X an (Spec K ) an (Spec K) an π 1 π 2
We then have that:

π 1 * (dd c (π * 1 φ 1 ) ∧ • • • ∧ dd c (π * 1 φ d )) = dd c φ 1 ∧ • • • ∧ dd c φ d .
An elegant way to encode information given by mixed Monge-Ampère operators is through the (metrized) Deligne pairing construction. it has a long history, starting from the complex case in Deligne's original article, treating the case of relative dimension 1 ( [Del]), further generalized by Elkik in [START_REF] Elkik | Fibrés d'intersections et intégrales de classes de Chern[END_REF], [START_REF] Elkik | Métriques sur les brés d'intersection[END_REF]. Its use to formulate functionals arising complex geometry has been popularized via [PRS], and recently, Deligne pairings have also been shown to be of great use in non-Archimedean geometry ([BHJ16], [BE], see also [START_REF] Dh Phong | Deligne pairings and the Knudsen-Mumford expansion[END_REF]Remark 6]). The non-Archimedean case over a point has been thoroughly developed in [BE]. This is the case which we review now.

Consider a (d + 1)-uple L 0 , . . . , L d of ample line bundles on X. To this data, we can associate a line bundle L 0 , . . . , L d on Spec K (i.e. a K-line), given by L 0 , . . . , L d = I⊂{0,...,d}

(-1) d+1-|I| d+1 j=1 (-1) j det H j i∈I L i ,
i.e. the top-iterated dierence of the determinant of cohomology. This construction has the property that it is symmetric and multilinear; furthermore, given a regular section s of L 0 , we have an isomorphism

L 0 , . . . , L d L 1 | div s , . . . , L d | div s .
By multilinearity and symmetry, the dierence L 0 , . . . , L d -M 0 , . . . , M d can be identied with the trivial line O X , . . . , O X on Spec K as soon as there exists some j with L j = M j . Recalling that (Spec K) an is a point, a non-Archimedean metric on the analytication of such a dierence can then be identied with a genuine real number. Consider now, for each i, a continuous psh metric φ i on L an i .

Theorem 2.4.1.2 ([BE, Theorem 8.16]). To the data above, one can associate a metric φ 0 , . . . , φ d on L 0 , . . . , L d an , which is uniquely characterized by the following properties: 1. the association of φ 0 , . . . , φ d to (φ 0 , . . . , φ d ) is symmetric and multilinear; 2. if s is a regular section of L 0 , then

φ 0 , . . . , φ d = φ 1 | div s , . . . , φ d | div s -ˆX log |s| φ 0 dd c φ 1 ∧ • • • ∧ dd c φ d .
Remark 2.4.1.3. In particular, one obtains the change of metric formula:

given another continuous psh metric φ 0 on L an 0 , we have

φ 0 , . . . , φ d -φ 0 , . . . , φ d = ˆX(φ 0 -φ 0 ) dd c φ 1 ∧ • • • ∧ dd c φ d .

The Monge-Ampère energy.

In the previous Section, we have seen that (continuous or bounded) psh metrics and bounded graded norms were closely related. It turns out that there are also natural objects mimicking the relative volumes and d 1 -distances on spaces of norms. Fix an ample line bundle L on X. We dene, given two continuous psh metrics φ 0 , φ 1 on L an , their relative Monge-Ampère energy:

E(φ 0 , φ 1 ) = 1 d + 1 d i=0 ˆX(φ 0 -φ 1 ) MA(φ (i) 0 , φ (d-i) 1
).

Note that this is always nite as the metrics are continuous. Of interest to us are the following properties: given φ 0 , φ 1 , φ 2 a triple of continuous psh metrics on L an , we have

• antisymmetry: E(φ 0 , φ 1 ) = -E(φ 1 , φ 0 ); • a cocycle property: E(φ 0 , φ 1 ) = E(φ 0 , φ 2 ) + E(φ 2 , φ 1 ); • increasingness in the rst argument: if φ 0 ≤ φ 1 , then E(φ 0 , φ 2 ) ≤ E(φ 1 , φ 2 ).
Remark 2.4.2.1. We would like to briey address the issue of conventions:

we follow those of [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF], wherein the Monge-Ampère energy is normalized by the volume of L. This is not the case in [BE].

The Monge-Ampère energy admits an extension to the class PSH(L an ) via

E(φ, φ ref ) = inf{E(ψ, φ ref ), ψ ≥ φ, ψ ∈ C 0 (L an ) ∩ PSH(L an )}
for a xed continuous psh metric φ ref .

We can also partially extend the relative Monge-Ampère energy, by setting

E(φ, φ ) = E(φ, φ ref ) -E(φ , φ ref )
for φ, φ ∈ PSH(L), which is dened whenever at least one of the two terms in the right-hand side is nite. This extended relative Monge-Ampère energy can therefore take -∞ or ∞ as values. This makes E continuous along decreasing nets.

Denition 2.4.2.2. The class E 1 (L an ) of nite-energy plurisubharmonic metrics is dened as the set of psh metrics φ on L an satisfying

E(φ, φ ref ) > -∞
for a reference continuous psh metric φ ref .

Due to the cocycle property of the energy, the class E 1 (L) is in fact independent of the choice of a reference metric, justifying our choice of notation for this class of metrics of nite energy.

Remark

Q-line bundle on X. Let ζ • , ζ • ∈ N • (L).
We then have:

lim m E(FS m (ζ m ), FS m (ζ m )) = vol(ζ • , ζ • ).
Conversely, given two continuous psh metrics φ and φ on L an , we have

E(φ, φ ) = vol(N • (φ), N • (φ )).
The rst part of the statement will be proven in the next Chapter. It will be helpful in proving that the asymptotic Fubini-Study operator is injective on the space of bounded graded norms modulo asymptotic equivalence.

2.4.3

The metric space of nite-energy psh metrics.

From now on, unless stated otherwise, we will assume that continuity of envelopes holds for (X, L). The space C 0 (L an ) ∩ PSH(L an ) can be endowed with a metric structure as follows:

Denition 2.4.3.1. Consider two metrics φ 0 , φ 1 ∈ C 0 (L) ∩ PSH(L). We dene

d 1 (φ 0 , φ 1 ) = d 1 (N • (φ 0 ), N • (φ 1 )),
where the distance in the right-hand side is the distance d 1 on bounded graded norms.

Remark 2.4.3.2. Dene for ease of notation vol(φ 0 , φ 1 ) = vol(N • (φ 0 ), N • (φ 1 )).

It follows (see e.g. [Reb20b, Remark 5.4.5]) that we have the formula d 1 (φ 0 , φ 1 ) = vol(φ 0 , P (φ 0 , φ 1 )) + vol(φ 1 , P (φ 0 , φ 1 )).

By the results from the previous Subsection, this is also equal to d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 )).

This distance is sometimes called the Darvas distance, as it was introduced in [Dar15] in the complex case. We will see that, as in [START_REF] Darvas | The Mabuchi geometry of nite energy classes[END_REF], it extends as a distance on the space of nite-energy metrics.

Proposition 2.4.3.3. The d 1 distance dened above is indeed a distance on the set of continuous psh metrics. Proof. Symmetry is immediate. The triangle inequality follows from taking the limit in the nite-dimensional triangle inequalities

k -1 d 1 (ζ k , ζ k ) ≤ k -1 d 1 (ζ k , ζ k ) + k -1 d 1 (ζ k , ζ k ) for any three bounded graded norms ζ • , ζ • , ζ • ∈ N • (R). If d 1 (φ, φ ) = 0, then N • (φ)
and N • (φ ) belong by denition to the same equivalence class of bounded graded norms. Since FS • • N • is the identity on continuous psh metrics and FS • factors through asymptotic equivalence, it follows that φ = φ . Finally, if φ = φ , then their distance is naturally zero.

We now turn to the case of nite-energy metrics.

Theorem 2.4.3.4. Assume continuity of envelopes to hold for (X, L). Then, (E 1 (L an ), d 1 ) is a metric space.

Remark 2.4.3.5. In [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], Boucksom-Jonsson prove completeness of (E 1 (L an ), d 1 )

in the trivially valued case, if and only if continuity of envelope holds. Their proof relies on deep results concerning spaces of measures of nite-energy on Berkovich spaces. The author expects to help in developing similar theory for the nontrivially-valued setting.

In order to prove the Theorem, we rst show that our distance is well-dened:

Proposition 2.4.3.6. Given two metrics φ 0 , φ 1 ∈ E 1 (L an ), P (φ 0 , φ 1 ) belongs to E 1 (L an ).

Proof. Fix a continuous L an -psh reference metric φ ref .

Let, for i = 0, 1, k → φ k i be sequences of continuous psh metrics decreasing to φ i . Assuming φ ref ≥ φ k 0 for all (large enough) k, we then have from Lemma 2.4.3.7 and the fact that the distance of two comparable metrics is a volume:

0 ≤ vol(P (φ k 0 , φ k 1 ), P (φ ref , φ k 1 )) = E(P (φ k 0 , φ k 1 ), P (φ ref , φ k 1 )) ≤ E(φ k 0 , φ ref ),
Since E and P are continuous along decreasing nets, this gives at the limit

0 ≤ E(P (φ 0 , φ 1 ), P (φ ref , φ 1 )) ≤ E(φ 0 , φ ref ) < ∞.
In particular, using the cocycle property, E(P (φ 0 , φ 1 ), φ ref ) is nite for any continuous psh reference metric, hence P (φ 0 , φ 1 ) ∈ E 1 (L an ).

The following Lemma was used in the proof of the previous Proposition.

Lemma 2.4.3.7. Let φ 0 , φ 1 be continuous L an -psh metrics. Then, for any continuous L an -psh metric φ, we have d 1 (P (φ 0 , φ), P (φ 1 , φ)) ≤ d 1 (φ 0 , φ 1 ).

Proof. This is essentially an asymptotic version of [BJ18a, Lemma 3.1]. By continuity of envelopes, the two metrics in the left-hand side are continuous (and psh), so that they dene bounded graded supnorms via the N • operator.

By [BE,Theorem 7.27],

P (φ 0 , φ) = FS • (N • (φ 0 ∧ φ)) = FS • (N • (φ 0 ) ∨ N • (φ))
(note that the statement of [BE,Theorem 7.27] uses the envelope Q which corresponds to the envelope dened by Fubini-Study metrics; but for continuous metrics, P = Q by [BE, Proposition 7.26]). Similarly,

P (φ 1 , φ) = FS • (N • (φ 1 ) ∨ N • (φ)), i.e. N • (P (φ 0 , φ)) = N • (φ 0 ) ∨ N • (φ)
and

N • (P (φ 1 , φ)) = N • (φ 1 ) ∨ N • (φ).
Chapter 3

The range of the asymptotic Fubini-Study operator over general non-Archimedean elds.

The main result.

In this Chapter, we will concern ourselves with proving the following result:

Theorem 3.0.0.1. Let X be a projective K-variety, endowed with a semiample line bundle L. Assume (X, L) to admit continuity of envelopes. The asymptotic Fubini-Study operator FS then denes a bijection:

FS : N • (L)/ ∼ → PSH ↑ (L).
This is a generalization of [BJ18a, Theorem 4.16], which treats the triviallyvalued case. The main ingredient in the proof is the following result:

Theorem 3.0.0.2. Assume (X, L) to admit continuity of envelopes. Then, given two bounded graded norms

ζ • , ζ • ∈ N • (L), we have that E(FS(ζ • ), FS(ζ • )) = vol(ζ • , ζ • ).
The idea of proof is as follows. In the case where FS(ζ • , which are graded norms on R(X, rL) generated in degree r by propagating ζ r through the surjective multiplication morphisms. This result builds on the theory of superadditive functions on Okounkov bodies developed in [START_REF] Chen | Distribution of logarithmic spectra of the equilibrium energy[END_REF].

3.1 Some preliminary results on approximations of bounded graded norms.

We begin with some preliminary results on bounded graded norms, focusing especially on nitely generated norms. As a convention, we will assume L to be a semiample Q-line bundle.

3.1.1 Bounded graded norms on section rings of semiample Q-line bundles.

Since L is a semiample Q-line bundle, there always exists a r such that R(X, rL) is generated in degree one. Thus, while norms generated in degree one may not necessarily exist (as they require by denition R(X, L) to be generated in degree one), the following class is always nonempty: Denition 3.1.1.1. We say that a graded norm ζ • on some subalgebra R(X, kL) of R(X, L) is nitely generated if it is generated in degree one on R(X, kL).

We may therefore extend the denition of a bounded graded norm to graded norms on R(X, L):

Denition 3.1.1.2. A bounded graded norm on R(X, L) is a graded norm ζ • on R(X, L
) such that there exists a nitely generated graded norm ζ • , generated in degree one on R(X, kL) for some k, such that

d ∞ (ζ k• , ζ • ) < ∞.
The proof of existence of a spectral measure , as we will show, still works in such generality, and we may similarly dene the d p distances.

(i) linear growth: Γ(L) is contained within a nitely generated monoid a 1 , . . . , a k , k < ∞, where for all i, a i ∈ {1} × N d ;

(ii) bigness: Γ(L) generates N d+1 as a group.

Denition 3.1.2.4. We will dene a convex body in R d to be a subset of

R
∆(L) = Cone(Γ(L)) ∩ ({1} × R d ).
This is the Okounkov body of Γ(L). Again, ∆(L) depends on the choice of ν.

Remark 3.1.2.6. This construction generalizes the moment polytope of a polarized toric variety, as in [LM09, Section 6.1].

We now prove a result which is similar in spirit to Fujita's approximation Theorem, which will be of use in the third Chapter. We rst start by recalling the following result: For all large enough integers m, we then have that:

K ∩ Γ m m = K ∩ Z d m ,
where Γ m is the m-th graded piece of Γ • .

We now prove the approximation result in question, which can also be seen as an ad hoc version of [Bou12, L1.21].

Lemma 3.1.2.8. Let Γ k

• be a sub-graded semigroup of some sub-graded semigroup Γ • of R d+1 , such that:

• Γ k 1 = Γ k ,
• Γ k r ⊆ Γ kr for all r ≥ 1,

• Γ • satises the properties (i) and (ii) of linear growth and bigness as in Section 4.1.

We then have that

k -d vol(∆(Γ k • )) → k→∞ vol(∆(Γ • )).
Proof. First remark that, by the inclusion property

Γ k r ⊆ Γ kr , we have that, for all k ≥ 1, ∆(Γ k • ) k ⊆ ∆(Γ • ).
If we can show that any compact (convex

) subset K of ∆(Γ • ) o is also included in ∆(Γ k • ) k
for large enough k, then our assertion would be true. Pick such a compact K, and embed it into another compact convex subset

L ⊂ ∆(Γ • ) o such that the number d(K, ∂L) = inf {d(x, ), x ∈ K, ∈ ∂L} is (strictly) positive. We then have compact inclusions K ⊂ L ⊂ ∆(Γ • ) o ,
with K not "touching" the boundary of L. By the bigness hypothesis, Γ • generates Z d+1 as a group. Then, the regularization of Γ k is Z d , whence, for all large enough k,

L ∩ Γ k k = L ∩ Z d k ,
(by Theorem 3.1.2.7), so that the convex hull of Γ k k naturally contains K. (It does not necessarily contain L.) Now, since

Γ k 1 = Γ k , the convex hull of Γ k k is contained in the scaled Okounkov body ∆(Γ k • ) k .
To conclude, we have a chain of compact inclusions

K ⊂ Hull Γ k k ⊂ ∆(Γ k • ) k ,
from which follows the desired inclusion of K.

Okounkov bodies and limit measures.

We now apply the constructions of the previous Subsection to a more precise geometric context.

Let x be a regular K-rational point of X, and pick a regular sequence (z 1 , . . . , z d ) in the local ring O X,x . By Cohen's structure Theorem, any element f ∈ O X,x may then be written as a formal power series

f = α∈N d f α z α ,
where the coecients f α belong to K. Pick a monomial valuation ν on K[[t 1 , . . . , t d ]]. Given a section s ∈ H 0 (X, L), one may pick a trivialization of L at x, so that s denes an element s x ∈ O X,x , and we proceed similarly for plurisections of L, allowing us to identify R(X, L) with a subalgebra of K[[t 1 , . . . , t d ]]. Note that this is independent of the choice of a trivialization.

We may now apply the constructions of the previous Subsection provided we have a good choice of a valuation with one-dimensional leaves.

We rst begin with the following denition.

Denition 3.1.3.1. A monomial order on N d is dened to be a total order ≤ satisfying the following properties:

1. given any α ∈ N d , 0 N d ≤ α;

2. given any α ∈ N d , for all α 0 , α 1 ∈ N d with α 0 ≤ α 1 , we have

α 0 + α ≤ α 1 + α.
Note that such an order naturally extends to Z d .

Denition 3.1.3.2. A monomial valuation or Gröbner valuation on R(X, L) is a valuation of the form

ν α∈Z d f α z α = min ≤ {α ∈ Z d , f α = 0},
with ≤ a monomial order on N d , and where we expand an element of the algebra R(X, L) as a power series as above. Since the transcendence degree of the residue eld of the valuation is 0, ν has one-dimensional leaves.

Since the leaves of a monomial valuation are one-dimensional, there exist elements s in each nonzero-dimensional gr k,α which can be expanded as

s = z α + β≥α v β z β , i.e.
Φ : Γ(L) → R k, α → -log [ ζ k,α (s k,α ) ] .
By submultiplicativity of ζ • and the fact that

s k,α • s ,β = s k+ ,α+β
in the algebra k∈N α∈Γ(H 0 (kL))

gr k,α (L), the function Φ so dened is then superadditive on the semigroup Γ(L). (2)

lim m→∞ m -1 d ∞ (ζ m , ζ m ) < ∞;
(3) there exists a uniform positive constant C such that

inf α∈Γ(H 0 (kL)) -Φ(k, α) ≥ -Cn,
where Φ is dened as above, and similarly for Φ dened using the graded norm ζ • .

The criteria (1) and (2) are by denition true. What remains is to prove

(3) which is equivalent to showing that (k, α) → Φ(k, α) is linearly bounded above in the rst variable, i.e. there exists a uniform positive constant C such that

Φ(k, •) ≤ C • k.
By denition of the quotient norm ζ k,α , it is enough to show that there exists

C such that -log ζ k (v) ≤ C • k
holds for all s of the form

s = z α + β≥α s β z β . (3.1)
By the nite growth property of Γ(L), we know that there exists a uniform positive constant C such that

α ∈ Γ(H 0 (kL)) ⇒ |α| ≤ C • k. (3.2)
Thus, it is enough to show that

-log ζ k (s) ≤ C(k + |α|) (3.3)
for all such s. We rst assume that ζ • = ζ •φ for some bounded metric φ on L. Now, we know that we can nd a trivialization τ x of L and analytic isomorphisms from a neighborhood U of a regular rational point x ∈

X to an open polydisc D = d 1 D(r i ) ⊂ K d , such that a section s ∈ H 0 (nL) satises log |s| nφ = log |s U | + n log |τ x | φ ,
for some analytic function s U of the form

s U (z) = z α + β≥α s β z β .
Since φ is bounded on U , so is the term n log |τ x | φ , and by the maximum principle, applied in each variable, we have that Since we no longer restrict ourselves to algebras generated in degree one, it should be interesting to understand the behaviour of the asymptotic spectral measure restricted to subalgebras. We have the following results:

r |α| ≤ sup U |s U |,
Proposition 3.1.3.4. Recall that, given any measure µ on the reals and any µ-measurable function f : R → R, f * µ denotes the pushforward of µ by f . Set

ζ • , ζ • ∈ N • (L).
We then have that:

• f (λ) = -λ ⇒ f * σ(ζ • , ζ • ) = σ(ζ • , ζ • ); • for any c ∈ R, f (λ) = λ + c ⇒ f * σ(ζ • , ζ • ) = σ(e -c ζ • , ζ • ); • for any r ∈ N * , f (λ) = rλ ⇒ f * σ(ζ • , ζ • ) = σ(ζ r• , ζ r• ), where ζ r• denotes the restriction of ζ • to the subalgebra R(X, rL).
This Proposition is the non-trivially valued equivalent of Propositions 3.4 and 3.5 of [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF], and are proven in the same manner.

Equidistribution of Okounkov points of superaddi-

tive functions associated to bounded graded norms.

We now prove a "norm-volume" version of the Fujita approximation Theorem, i.e. that the asymptotic relative volumes between two bounded graded norms can be approximated using the asymptotic relative volumes between their nitely generated approximations. We begin by recalling the following Proposition:

Proposition 3.1.4.1 ([CM15, L4.1, T4.3]). Assume Φ is a superadditive function

Φ : Γ(L) → R, such that Φ(0, 0 N d ) = 0. For any t ∈ R, set Γ Φ,≥t = {(n, α) ∈ Γ(L), Φ(n, α) ≥ n • t}.
Then, Γ Φ,≥t is a sub-semigroup of Γ(L) satisfying properties (i)-(ii) of Lemma 3.1.2.3 whenever

t < θ = lim n→∞ sup α∈Γn(L) n -1 Φ(n, α). Remark 3.1.4.2. It is immediate that l < t ⇒ Γ Φ,≥l ⊆ Γ Φ,≥t .
Denition 3.1.4.3. Let Φ be a superadditive function on Γ(L). We set

G Φ : ∆(Γ(L)) → R ∪ {-∞}, (n, α) → sup{t ∈ R ∪ {-∞}, (n, α) ∈ ∆(Γ Φ,≥t )}.
The function G Φ is the Chebyshev function of the semigroup Γ(L) (associated to Φ). The term concave transform is also common in the literature, see e.g. [START_REF] Witt | Transforming metrics on a line bundle to the Okounkov body[END_REF] and [START_REF] Küronya | Concave transforms of ltrations and rationality of Seshadri constants[END_REF].

Remark 3.1.4.4. By [START_REF] Boucksom | Okounkov bodies of ltered linear series[END_REF] this function is concave, hence continuous, on the interior of ∆(Γ(L)).

The proof of our main result relies on the following equidistribution Theorem for the values of a superadditive function dened on an Okounkov body: Theorem 3.1.4.5 ([CM15, T4.3, R4.4]). Let Φ be a superadditive function from Γ(L) to R, whose lim sup we denote by θ as previously. Let µ(k) be the nitely supported probability measure on R dened as

µ(k) = α∈Γ k (L) δ k -1 Φ(k,α) .
This sequence then converges to a compactly supported probability measure µ on R satisfying

µ([t, ∞)) = vol(∆(Γ Φ,≥t • )) vol(∆(Γ • )) ,
for any t ≤ θ. Furthermore, µ is equal to the pushforward of the normalized Lebesgue measure on the Okounkov body ∆(Γ(L)) by the Chebyshev function

G Φ .
If Φ is a superadditive function dened on an Okounkov body, associated to a bounded graded norm ζ • as before, we denote the limit measure obtained Proposition 3.1.4.6. Let L be such that R(X, L) is generated in degree one. Let ζ • be a bounded graded norm on R(X, L). Consider, for each k ∈ N * , the bounded graded norm ζ (k)

• on R(X, kL) generated in degree one by ζ k , i.e. the sequence of quotient norms induced by ζ k and the symmetry morphisms

H 0 (kL) r H 0 (rkL)
for all r ∈ N * . Set

Γ(kL) = {(n, α) ∈ Γ(L), k|n}.
We then have

µ(ζ (k) • ) k→∞ µ(ζ • ),
where denotes weak convergence of measures, in particular: the sequence of functions

t → ´t -∞ dµ(ζ (k) • ) converges pointwise to t → ´t -∞ dµ(ζ • ).
Proof. Let Φ and Φ k be the superadditive functions associated to the norms

ζ • and ζ (k)
• . We rst notice the following properties of Φ and the Φ k :

(i) Φ k (k, α) = Φ(k, α), for all (k, α) ∈ Γ k ; (ii) Φ k (kn, α) ≤ Φ(kn, α), for all (kn, α) ∈ Γ k• ; (iii) if d|k, then Φ d (kn, α) ≤ Φ k (kn, α), for all (kn, α) ∈ Γ k• .
Let θ k and θ be the above bounds on the supports of the appropriate measures. We then show that

µ(ζ (k) • )([t, θ]) → µ(ζ • )([t, θ]), for all t ∈ [-∞, θ]. Now, since µ(ζ (k) • )([t, θ]) = vol ∆(Γ Φ k ,≥t k• ) vol(∆(Γ k• )) , and vol(∆(Γ k• )) -1 = vol(∆(Γ • )) -1 ,
the problem reduces to showing that the sequence of functions (v k ) k , dened as

v k : t → vol ∆(Γ Φ k ,≥t k• ) converges pointwise to v : t → vol(∆(Γ Φ,≥t •
)).

Note that (ii), (iii), and the expressions

θ = lim n→∞ sup Γn Φ(n, α) n < ∞,
and

θ k = lim n→∞ sup (kn,α)∈Γ kn Φ k (kn, α) n < ∞
imply that (θ k ) k is an increasing sequence converging to θ.

Finally, the semigroups Γ

Φ (k!) ,≥t (k)•
and Γ Φ,≥t

•

, satisfy the hypotheses of Lemma 3.1.2.8 (note (i)), which yields

v k (t) → v(t),
concluding the proof.

We may now prove our desired Theorem.

vol(ζ (k) • , ζ • (k) ) → k→∞ vol(ζ • , ζ • ).
Proof. Let Φ and for all k, Φ k be the superadditive functions associated to

the norms ζ • and ζ (k) • ) respectively.
Recall the identity

vol(ζ • , ζ • ) = lim m→∞ m -1 vol(ζ • , ζ • ). Note that ˆR λ dµ(ζ m ) -ˆR λ dµ(ζ m ) = m -1 α∈Γm(L) [Φ(m, α) -Φ (m, α)] ,
where µ(ζ m ) and µ(ζ m ) are dened as the nitely supported measures as in Theorem 3.1.4.5. By [START_REF] Chen | Distribution of logarithmic spectra of the equilibrium energy[END_REF](29)], the quantity on the right is identied with

m -1 vol(ζ m , ζ m ), so that at the limit, ˆR λ dµ(ζ • ) -ˆR λ dµ(ζ • ) = vol(ζ • , ζ • ).
Doing the same process with ζ

(k) • and ζ (k)
• , we then nd that

ˆR λ dµ(ζ (k) • ) -ˆR λ dµ(ζ (k) • ) = vol(ζ (k) • , ζ • (k)
).

An application of Theorem 3.1.4.6 then yields the desired convergence.

3.1.5 Approximation of graded norms generated in degree one via graded norms coming from models.

It is important to emphasize that bounded graded norms ζ • generated in degree one are very easy to study: their asymptotic behaviour is heavily controlled by that of ζ 1 and of the asymptotic structure of the underlying algebra, as one can see from the following result:

Lemma 3.1.5.1. Let V • be a graded K-algebra generated in degree one. Let

ζ • , ζ • ∈ N • (V • )
be generated in degree one. We then have that:

d ∞ (ζ • , ζ • ) = d ∞ (ζ 1 , ζ 1 ).
Proof. This follows on repeatedly applying Proposition 1.2.3.10. Set a =

d ∞ (ζ 1 , ζ 1 ). For any m > 1, we have that φ m : V m 1 → V m is surjective. Consider v ∈ V m , and lifts ṽ of v in V m 1 , which themselves lift to ṽ ∈ V ⊗m 1
. We naturally have that e -ma (ζ 1 ) ⊗m ( ṽ) ≤ (ζ 1 ) ⊗m ( ṽ) ≤ e ma (ζ 1 ) ⊗m ( ṽ), so that, applying the above Proposition, e -ma (ζ 1 ) m (ṽ) ≤ (ζ 1 ) m (ṽ) ≤ e ma (ζ 1 ) m (ṽ),

and nally, since a graded norm generated in degree one is a quotient,

e -ma (ζ m )(s) ≤ (ζ m )(v) ≤ e ma (ζ v )(s).
This establishes

d ∞ (ζ • , ζ • ) ≤ d ∞ (ζ 1 , ζ 1 ),
and since the d ∞ distance is dened as a sup, we in fact have equality.

The main result of this Subsection will then be a powerful approximation Theorem for graded norms generated in degree one. It requires specic constructions of certain bounded graded norms, which cannot in general be assumed to be being generated in degree one; however, they will coincide in all high enough degrees with one such norm. Hence, we introduce the following denition, to make our later statements lighter.

Denition 3.1.5.2. We say that a bounded graded norm ζ • is eventually generated in degree one if there exists a norm generated in degree one

ζ o
• , and a positive integer r, such that for all m ≥ r,

ζ m = ζ o m .
We will say that ζ • eventually coincides with ζ o

• .

Let L be an ample line bundle on X whose algebra of sections is generated in degree one. We now describe how to construct, starting from a lattice norm on H 0 (L), a model (X , L) of (X, L), such that the bounded graded norm associated to the sections of L is eventually generated in degree one.

Let ζ thus be a lattice norm on H 0 (L), which as we recall means that there exists a basis of sections (s i ) of H 0 (L) which is orthonormal for ζ. Denote V 1 the K • -submodule of H 0 (L) generated by this basis of sections, i.e. the unit ball of ζ. Then, the surjective symmetry morphisms φ r :

H 0 (L) r → H 0 (rL) of R(X, L) being surjective for all r ≥ 1, V 1 induces a K • -subalgebra V • of R(X, L)
, which is furthermore generated in degree one, and torsion-free. The scheme

X = Proj V • (3.4)
is then at and projective over K • . Let L be its twisting sheaf O X (1). (X , L) is a model of (X, L). Furthermore, for all m large enough, H 0 (mL) coincides with V m (see [Har77, Ex. II-5.14]). In particular, the sequence of norms

(ζ H 0 (mL) ) m
is eventually generated in degree one, and the norm generated in degree one with which it eventually coincides is generated by ζ.

We may then prove the following result:

Proposition 3.1.5.3. Assume K to be densely valued, and let ζ • be generated in degree one. Then, for all ε > 0, there exists a model

(X ε , L ε ) of (X, L), such that, for large enough m, d ∞ (ζ m , ζ H 0 (mL ε ) ) < mε.
Proof. Since K is densely valued, for all ε > 0, there exists a lattice norm

ζ ε with d ∞ (ζ 1 , ζ ε ) < ε.
Being a lattice norm, we associate to ζ ε a model (X ε , L ε ) as in the construction (3.4) above, whose associated graded norm ζ H 0 (•L ε ) eventually coincides with the norm generated in degree one by ζ ε .

Since ζ • and ζ H 0 (•L ε ) are both eventually generated in degree one, we have that

d ∞ (ζ m , ζ H 0 (mL ε ) ) ≤ md ∞ (ζ 1 , ζ ε ) < mε
for all m large enough.

Finally, we may prove the main Theorem of this section.

Theorem 3.1.5.4. Assume K to be densely valued, and assume L to be such that R(X, L) is generated in degree one. Let ζ • , ζ • be bounded graded norms generated in degree one on L. Then, for all ε > 0, there exist models (X ε , L ε ) and (Y ε , M ε ) of (X, L), such that:

vol(ζ H 0 (•L ε ) , ζ H 0 (•M ε ) ) → ε→0 vol(ζ • , ζ • ).
Proof. We pick sequences of models (X ε , L ε ) and (Y ε , M ε ) of (X, L) as in Proposition 3.1.5.3. Using the cocycle condition on volumes, we have that

vol(ζ • , ζ • ) = vol(ζ • , ζ H 0 (•L ε ) ) + vol(ζ H 0 (•L ε ) , ζ H 0 (•M ε ) ) + vol(ζ H 0 (•M ε ) , ζ • ), so that it is then enough to prove vol(ζ • , ζ H 0 (•L ε ) ) → ε→0 0.
(The proof for L being also valid for M.) Since volumes respect a Lipschitz property with respect to the d ∞ -distance (Proposition 1.2.3.9), we have

| vol(ζ • , ζ H 0 (•L ε ) )| = | vol(ζ • , ζ H 0 (•L ε ) )-vol(ζ • , ζ • )| ≤ lim sup m m -1 d ∞ (ζ m , ζ H 0 (mL ε ) ).
In light of Proposition 3.1.5.3, we then have that

| vol(ζ • , ζ H 0 (•L ε ) )| ≤ d ∞ (ζ 1 , ζ H 0 (L) ) < ε,
which concludes the proof.

3.2

The range of the asymptotic Fubini-Study operator.

3.2.1 Relating asymptotic volumes and Monge-Ampère energies.

The goal of this Subsection is to prove the following Theorem, a generalization of [BJ18a, T4.13], where we consider a general non-Archimedean eld, rather than one which is trivially valued.

Theorem 3.2.1.1. Let L be a semiample Q-line bundle on a projective K-

variety X. Let ζ • , ζ • ∈ N • (L).
We then have:

lim m E(FS m (ζ m ), FS m (ζ m )) = vol(ζ • , ζ • ).
As a rst reduction, note that we can assume L to be a globally generated (genuine) line bundle and the algebra of sections to be generated in degree one, thanks to Proposition 3.1.3.4.

We now show that we can reduce to the case where K is a maximally complete, algebraically closed, densely valued eld, as in Proposition 1.1.3.4.

Lemma 3.2.1.2. Assume K to be any non-trivially valued, non-Archimedean eld, and that Theorem 3.2.1.1 holds for the base change of X K to an algebraically closed extension K of K. Then, Theorem B holds for X.

Proof. By invariance of nite-dimensional volumes under ground eld extension, the right-hand side is indeed invariant under ground eld extension, so that we only have to take care of the energy side of the equation. Consider the base change X K and its pullback line bundle L K . Note that the ground eld extension R(X, L) K of the algebra of sections of L coincides with R(X K , L K ). Consider the associated norms ζ •,K and ζ •,K .

• by Proposition 1.2.2.4, the Fubini-Study operators associated to each individual norm coincide with those associated to their ground eld extension, and that (say)

FS m (ζ m,K ) = π 1 * FS m (ζ m );
• by Proposition 2.4.1.1,

π 1 * MA(FS m (ζ m,K ), FS m (ζ m,K )) = MA(FS m (ζ m ), FS m (ζ m )),
where MA(φ, φ ) denotes any mixed Monge-Ampère measure involving only φ and φ .

It follows that both quantities in the assertion of Theorem B are invariant under ground eld extension. Using that the Theorem then holds over X K , this nishes the proof.

From now on, assume K to be as in Proposition 1.1.3.4.

Lemma 3.2.1.3. Theorem 3.2.1.1 holds whenever ζ • and ζ • are both graded norms generated in degree one.

Proof. Pick approximations ζ H 0 (•L ε ) and ζ H 0 (•M ε ) as in Theorem 3.1.5.4. By Lemma 3.2.1.4 below, we have, for all ε > 0,

E(φ L ε , φ M ε ) = vol(ζ H 0 (•L ε ) , ζ H 0 (•M ε ) ).
Now, the statement of Theorem 3.1.5.4 is that

vol(ζ H 0 (•L ε ) , ζ H 0 (•M ε ) ) → ε→0 vol(ζ • , ζ • ).
In particular, by construction, we have that

lim m FS m (ζ H 0 (mL ε ) ) = FS 1 (ζ H 0 (L ε ) ) = φ L ε ,
so that the Lemma is proven once we show that

lim m E(FS m (ζ H 0 (mL ε ) ), FS m (ζ H 0 (mM ε ) )) → ε→0 lim m E(FS m (ζ m ), FS m (ζ m )).
But using the 1-Lipschitz property of the operator FS m with respect to the sup norm of metrics and the d ∞ -distance, we nd that for all m, for all ε > 0, sup

X |FS m (ζ H 0 (mL ε ) ) -FS m (ζ m )| ≤ ε, so that nally, lim m FS m (ζ H 0 (mL ε ) ) → ε→0 lim m FS m (ζ m ),
uniformly. Proceeding similarly for M, and then using continuity of the Monge-Ampère energy along uniform limits, we nd the desired result.

We then have the following Lemma, as promised.

Lemma 3.2.1.4. Assume (X , L), (Y, M) to be semiample models of L dened on the same model X of X. Denoting φ L and φ M their associated model metrics, we then have that

E(φ L , φ M ) = vol(ζ H 0 (•L) , ζ H 0 (•M) ).
Proof. We rst start by stating the following equality ([BE, L9.17]):

vol(ζ H 0 (•L) , ζ H 0 (•M) ) = vol(N • (φ L ), N • (φ M )).
Note that our conventions for the volume and energy are dierent from those of [BE], but as

lim h 0 (mL) m dim X = vol(L),
the changes cancel out. Furthermore, their notation vol(L, φ, ψ)

corresponds to vol(N • (φ), N • (ψ))
in our case.

The above equality follows from earlier results of [BE], wherein it is shown that

d ∞ (ζ H 0 (mL) , N m (φ L )) = O(1),
([BE, T6.4]) so that Lipschitz continuity of the volume with respect to the d ∞ -distance concludes. Then, the Lemma is proven by applying Theorem 9.15 of [BE] to φ L and φ M .

We now prove the main Theorem.

Proof. Assume now that both norms are not necessarily nitely generated.

By surjectivity of H 0 (kL) m → H 0 (kmL) for large and divisible enough k, m > 0, we may endow each H 0 (kmL) with the quotient norm induced by this morphism using the norms ζ k , ζ k . We denote these norms ζ

(k) m , ζ (k) m .
These dene graded norms, generated in degree one, on R(X, kL). Consider their associated Fubini-Study metrics:

FS k (ζ (k) • )
and

FS k (ζ (k)
• ). Recall that the Theorem holds for those norms. Now, since (FS k (ζ

(k) • )) k , resp. (FS k (ζ (k)
• )) k are decreasing nets, by continuity of E along decreasing nets follows:

lim k→∞ E FS k (ζ (k) • ), FS k (ζ (k) • ) = E(lim k FS k (ζ k ), FS k (ζ k )).
The right-hand side limit, that is,

lim k→∞ vol(ζ (k) • , ζ (k)
• ) = vol(ζ • , ζ • ),
is the statement of Theorem 3.1.4.7.

3.2.2

The asymptotic Fubini-Study operator descends to a bijection.

We now prove the following Theorem, which is a generalization of [BJ18a, T4.16] to the nontrivially-valued case:

Remark 3.2.2.4. Note that the previous proof shows that there is an expression of the d 1 distance using Monge-Ampère energies, analogous to [BJ18a, C4.21]:

d 1 (ζ • , ζ • ) = E(FS(ζ • ), Q(FS(ζ • ) ∧ FS(ζ • ))) + E(FS(ζ • ), Q(FS(ζ • ) ∧ FS(ζ • ))).
Remark 3.2.2.5. One can see the asymptotic Fubini-Study operator as giving an injective isometry with dense image of the space of nitely-generated graded norms modulo asymptotic equivalence, into the metric space E 1 (L).

Since it is not known whether the envelope P coincides with the envelope Q for general metrics approachable from below, such a result does not hold a priori for the entire space of bounded graded norm modulo asymptotic equivalence, although it is conjectured that this is true.

5. the Monge-Ampère energy is ane along φ t , and it is the unique psh segment joining φ 0 and φ 1 with this property. If the endpoints are continuous, this maximal psh segment is furthermore continuous in time and space.

4.1 Plurisubharmonic segments in non-Archimedean geometry.

4.1.1 Fubini-Study segments, plurisubharmonic segments.

The basic building block for our plurisubharmonic segments are what we call Fubini-Study segments, which we dene as follows.

Denition 4.1.1.1. A Fubini-Study segment is a map

[0, 1] t → φ t ∈ H(L)
such that there exist a nite basepoint-free collection of sections (s i ) of some H 0 (kL), and for each i, real constants λ i and λ i ∈ R such that for all t,

φ t = k -1 max i log |s i | + (1 -t)λ i + tλ i .
Note the similarity with our denition of Fubini-Study metrics. Again, such segments are immediately seen to be convex in t, stable under nite maxima and addition of constants.

Remark 4.1.1.2. In particular, the image by the operator FS k of some norm geodesic t → ζ t in H 0 (X, kL), with ζ 0,1 diagonalizable, denes a Fubini-Study segment: indeed, given a basis s = (s i ) i codiagonalizing the endpoints, we have for all t, i

ζ t (s i ) = ζ 0 (s i ) 1-t ζ 1 (s i ) t so that F S k (ζ t ) = max i (log |s i | -(1 -t) log ζ 0 (s i ) -t log ζ 1 (s i )) .
Then, following the idea that psh metrics are decreasing limits of Fubini-Study metrics, we dene Finally, we show that our segments also satisfy the remaining properties of Proposition 2.3.1.6.

Proposition 4.1.1.5. Plurisubharmonic segments satisfy the following properties:

1. the convex combination of two psh segments is a psh segment; 2. the addition of a L-psh segment and a M -psh segment is a L + M -psh segment; 3. if a net of psh segments converges uniformly to a limit segment, then this limit segment is psh. Proof. We start with (2). The statement follows from the case of Fubini-Study segments. Consider thus two such segments

t → φ t = k -1 max i log |s i | + (1 -t)λ i + tλ i and t → ψ t = -1 max j log |t j | + (1 -t)γ j + tγ j . Then φ t + ψ t = (k -1 )(max i (log |s i | + (1 -t) λ i + t λ i ) + max j (log |t k j | + (1 -t)kγ j + tkγ j )) = (k -1 )(max i,j log |s i t k j | + (1 -t)( λ i + kγ j ) + t( λ i + kγ j )),
which is a Fubini-Study segment on (k )(M + L).

The third point follows from noticing that we can use sequences rather than nets when dealing with uniform convergence, and then adding constants to reduce to the case of a decreasing limit of psh segments, which converges by denition to a psh segment. Finally, the rst point follows again from the Fubini-Study case, from a simple computation similar to the proof of (2).

Given a convex function f : R p × R q → R ∪ {∞}, it is well-known that the inmum of the marginals

R q y → inf x∈R p f (x, y)
is also convex. This generalizes in multiple ways, as Prekopa's theorem ( [START_REF] Prékopa | On logarithmic concave measures and functions[END_REF]), but also to plurisubharmonic functions (independent of the imaginary part of the variable over which the inmum is taken). This is the well-known Kiselman minimum principle ([Kis78], [START_REF] Christer O Kiselman | Plurisubharmonic functions and their singularities[END_REF]), and a crucial tool in the study of plurisubharmonic functions. We propose here a non-Archimedean version of this result.

Lemma 4.2.2.1 (Non-Archimedean Kiselman minimum principle). Let [0, 1] t → φ t be a psh segment in PSH(L). Then, for each τ ∈ R, the Legendre

transform φτ : x → inf t∈[0,1] φ t (x) -tτ is in PSH(L)
. Furthermore, by Legendre duality,

φ t = sup τ ∈R φτ + tτ.
Proof. Since a psh segment on L is a global decreasing limit of Fubini-Study segments, and one notices the map between segments of psh metrics

(t → φ t ) → (τ → φτ )
to be continuous along decreasing sequences of segments (by its denition as an inmum over t of psh metrics), it is enough to consider the case where t → φ t is a Fubini-Study segment, i.e. there exists a nite basepoint-free collection of sections (s i ) i∈I of H 0 (kL) for some k such that

φ t = max i∈I (log |s i | + tλ i + c i ),
with xed constants (λ i ) and (c i ).

Set τ ∈ R, and consider the functions

f : [0, 1] × R |I| → R, (t, s) → k -1 max i s + t(λ i -τ ) + c i and g : R |I| → R, s → inf t∈[0,1] f (t, s).
It is clear that φτ is the composition of g and the formal tropicalization map trop :

X x → log |s 1 |(x), . . . , log s |I| (x) ∈ (R ∪ {-∞}) |I| .
We rst show that g is a piecewise-linear convex map, and we will explain how from this result we can prove that φτ is Fubini-Study.

The strict epigraph of g

E g = {(s, y) ∈ R |I| × R, g(s) < y}
is the image under the projection p :

[0, 1] × R |I| × R → R |I| × R of the epigraph of f E f = {(t, s, y) ∈ [0, 1] × R |I| × R, f (t, s) < y}.
One notices that, since f is piecewise-linear and convex in all variables, E f is convex and its closure is a piecewise-linear set. Since both of those properties are preserved under linear maps, E g is also convex and PL. This implies that the same holds for the function g. We will show this with k = 1 for clarity, and all the arguments below can be adapted for general k upon dividing where needed. Indeed, since f is convex, PL, and satises the property above, there exist nitely many ane functions f j such that f = max j f j and f j (z 1 , . . . , z p ) = i α i,j z i + b with i a i,j = 1. Since a maximum of psh metrics is psh, it is enough to prove that f j (log |s 1 |, . . . , log |s p |) is psh. We will therefore drop the subscript j and write a i for the coecients above. Now, the monotonicity condition ensures that the a i all belong to [0, 1], i.e. the vector (a i ) i is in the p-dimensional simplex. We assume at rst that

a i = p i /q i ∈ Q ∩ [0, 1]. Denote α i = p i • q -1 i • j q j
and remark that, by the simplex condition,

i α i = i q i .
Then,

f (log |s 1 |, . . . , log |s p |) = b + i a i log |s i | = b • i q i i q i + 1 i q i log i |s α i i |. Now, i s α i i is a section of ( i α i )L = ( i q i )L.
Therefore, f is in the image of F S i q i , i.e. it is a Fubini-Study metric. If some of the coecients are irrational, then f can be uniformly approximated by a function with rational coecients satisfying all the conditions above, i.e. f (log |s 1 |, . . . , log |s p |) can be uniformly approximated by Fubini-Study metrics, which shows that it is psh.

• for large negative τ , by boundedness again we have φ 0 ≤ φ 1 -τ so that tτ + P (φ 0 , φ 1 -τ ) = tτ + P (φ 0 ) ≤ P (φ 0 ).

Therefore, for some constant C(S) > 0, and for all t ∈ S,

φ t = sup τ ∈[-C(S),C(S)]
tτ + φτ , a supremum of continuous functions over a compact set, which is therefore continuous. We have proven that (t, x) → φ t (x) is continuous on (0, 1) × X.

We now prove that it is continuous up to the boundary. We start with the case t = 0. For very small values of t, very positive values of τ will never contribute to the supremum, so that we need only consider values of τ bounded above by some constant C 0 . Since we always have

P (φ 0 , φ 1 -τ ) ≤ P (φ 0 ) = φ 0 ,
it follows that for very small values of t, tτ + φτ ≤ tC 0 + P (φ 0 , φ 1 -τ ) ≤ tC 0 + φ 0 .

Taking the supremum, we thus have φ t -φ 0 ≤ tC 0 .

(4.1)

By boundedness of φ 1 , there exists a negative enough value of τ , say C 0 , such that φC 0 = P (φ 0 ) = φ 0 , i.e. for all small enough t, there exist some τ with

C 0 t + φ 0 ≤ tτ + φτ which implies C 0 t ≤ φ t -φ 0 .
Combining this with (4.1) shows that φ t converges uniformly to φ 0 for small enough t, which proves continuity at t = 0. If t is very close to 1, the argument proceeds in the same way, by noticing that P (φ 0 , φ 1 -τ ) = P (φ 0 + τ, φ 1 ) -τ.

134

To show that t → φ t is a psh segment, we consider the net of psh segments

I → max τ ∈I (t → tτ + φτ ),
where I belongs to the set of nite collections of elements in R, directed by inclusion. This does indeed dene a psh segment for all such I, as a nite maximum of psh segments. By denition, the limit of this net is t → φ t , and it is naturally increasing along inclusion. By Dini's Theorem, this gives a sequence of psh segments converging uniformly to t → φ t , which is equivalent to saying that it is a psh segment, proving our result.

Lemma 4.2.3.2. The curve

t → φ t = sup τ ∈R tτ + φτ
is the largest psh segment joining φ 0 and φ 1 .

Proof. We rst show that φ t bounds from above all psh segments between the endpoints. Plurisubharmonic segments are dened as decreasing limits of Fubini-Study segments. Therefore, at the endpoints, Dini's theorem gives uniform convergence, ensuring that if k → ψ k t is a sequence of Fubini-Study segments decreasing to a psh segment ψ t with ψ 0,1 ≤ φ 0,1 , then we can assume that for all large enough k, ψ k 0,1 ≤ φ 0,1 . Therefore, it is enough to treat the case of Fubini-Study segments.

Consider therefore a Fubini-Study segment t → ψ t with ψ 0 ≤ φ 0 ,

ψ 1 ≤ φ 1 . Let R τ → ψτ = inf t∈[0,1]
ψ t -tτ be its Legendre transform. By the minimum principle Lemma 4.2.2.1, ψτ ∈ H(L) for all τ . Taking t = 0, 1 we have

ψτ ≤ ψ 0 ≤ φ 0 and ψτ ≤ ψ 1 -τ ≤ φ 1 -τ.
As ψτ is psh, we then have ψτ ≤ φτ = P (φ 0 , φ 1 -τ ).

Taking the Legendre transform again, we nd

ψ t = sup τ ∈R ψτ + tτ ≤ sup τ ∈R φτ + tτ = φ t ,
which establishes our rst desired result: φ t bounds all psh segments by above. By Lemma 4.2.3.1, φ t is itself a psh segment, which concludes the proof.

We then have all the tools in hand to prove the Theorem. We now turn to the statements (3) and (4) of Theorem 4.2.1.1. From our denition of the maximal psh segment as a Perron envelope, it is not obvious how to recover the desired properties. Instead, we will obtain a "quantized" characterization of that segment, using sequences of Fubini-Study segments.

Let φ 0 , φ 1 be two continuous psh metrics as before. To those metrics, we can associate the bounded graded norms N • (φ i ), i = 0, 1.

Φ t : t → lim k FS k (ζ t k )
exists for all t by Fekete's lemma. By the same lemma, this is in fact a supremum over k. We claim that this limit coincides with the maximal psh segment φ t joining φ 0 and φ 1 . To that end, we show that Φ t bounds all psh segments by above.

Proposition 4.2.4.1. Let ψ t be a plurisubharmonic segment joining two metrics ψ 0 ≤ φ 0 and ψ 1 ≤ φ 1 in C 0 (L) ∩ PSH(L). We then have that

ψ t ≤ Φ t for all t ∈ [0, 1].
Proof. As in the proof of Lemma 4.2.3.2, Dini's theorem gives uniform convergence of a sequence k → ψ k t of Fubini-Study segments decreasing to ψ t , so that for all large enough k, ψ k 0,1 ≤ φ 0,1 , since we have assumed ψ 0,1 ≤ φ 0,1 .

Therefore, it enough to prove the result for all Fubini-Study segments ψ t with ψ 0,1 ≤ φ 0,1 . The argument is similar to that of [DLR20, Proposition 2.12].

We start by xing some notation. As we have just said, we can assume t → ψ t to be a Fubini-Study segment in the image of some FS k , with ψ 0 ≤ φ 0 , ψ 1 ≤ φ 1 . Denote by:

• t → ψt the image by FS k of the norm geodesic in H 0 (kL) joining N k (ψ 0 ) and N k (ψ 1 );

• t → Φ k t the image by FS k of the norm geodesic in H 0 (kL) joining N k (φ 0 ) and N k (φ 1 ).

By denition, since Φ t = sup m Φ m t , we have

Φ k t ≤ Φ t . (4.2)
Since ψ 0 = ψ0 , ψ 1 = ψ1 , by the maximum principle for norm geodesics Lemma 4.1.2.1, we have ψ t ≤ ψt . as well as

Φ k 0 = FS k (N k (φ 0 )) and Φ k 1 = FS k (N k (φ 1 )),
we then have ψ0 ≤ Φ k 0 and ψ1 ≤ Φ k 1 , so that, by monotonicity of norm geodesics in the form of Proposition 1.2.5.4

(applied to the N k (ψ 0,1 ) ≥ N k (φ 0,1 )), we have ψt ≤ Φ k t .

(4.4)

Combining (4.3), (4.4), and (4.2), we nally have

ψ t ≤ Φ t ,
as desired.

Theorem 4.2.4.2. Let φ 0 , φ 1 be two continuous psh metrics. The segment

Φ t : t → lim k FS k (ζ t k ),
where t → ζ t k is the norm geodesic joining N k (φ 0 ) and N k (φ 1 ), coincides with the maximal psh segment φ t joining φ 0 and φ 1 . Proof. By Proposition 4.2.4.1, we have Φ t ≥ φ t , since in particular φ t is a psh segment joining φ 0 and φ 1 . But, by Fekete's lemma, the limit of the sequence k → FS k (ζ t k ) is in fact a supremum, i.e. Φ t is a supremum of a subset of the set of psh segments below φ 0 and φ 1 . This gives Φ t ≤ φ t , which proves our result. 4.2.5 Proof of Theorem 4.2.1.1, (3) and (4).

Using our newfound expression for φ t , we may now nish the proof of the Theorem.

Proof of Theorem 4.2.1.1 (3)-(4). We start with (4). By the cocycle property, we can set φ ref = φ 1 . By [BE,Theorem 9.15], E(φ t , φ 1 ) = vol(φ t , φ 1 ).

For any k, let ζ t k be the norm geodesic joining N k (φ 0 ) and N k (φ 1 ), and write

vol(ζ t k , ζ 1 k ) = h 0 (kL) -1 λ i,k (ζ t k , ζ 1 k ) = (1 -t) h 0 (kL) -1 λ i,k (ζ 0 k , ζ 1 k ) = (1 -t) vol(ζ 0 k , ζ 1 k ).
Taking the limit, we have

vol(ζ t • , ζ 1 • ) = vol(φ t , φ 1 ) = (1 -t) vol(φ 0 , φ 1 )
.

By [BE] again, this is equal to the energy:

E(φ t , φ 1 ) = vol(φ t , φ 1 ).

vol(ζ t k,ε , ζ 1 k,ε ) = (1 -t) vol(ζ 0 k,ε , ζ 1 k,ε ) so that at the limit vol(ζ t k , ζ 1 k ) = (1 -t) vol(ζ 0 k , ζ 1 k )
still holds for all k, proving our result.

We now show that such a psh segment is unique. Fix a reference metric φ ref ∈ C 0 (L) ∩ PSH(L). Assume t → ψ t is another such segment joining two metrics φ 0 , φ 1 ∈ C 0 (L) ∩ PSH(L), i.e. it is a psh segment along which the energy is ane. By the maximum principle Theorem 4.1.3.1, we then have ψ t ≤ φ t for all t, and t → E(φ t , φ ref ), t → E(ψ t , φ ref ) are then ane functions with the same endpoints, hence for all t E(φ t , φ ref ) = E(ψ t , φ ref ).

By Proposition 4.2.6.2, since ψ t ≤ φ t and the energies coincide, we have ψ t = φ t .

Finally, for (3), the same idea as in (4) works: for all k, and for all t and t in [0, 1] we nd

d 1 (ζ t k , ζ t k ) = |t -t |d 1 (ζ 0 k , ζ 1 k ),
and we conclude by passing to the limit.

Remark 4.2.5.1. However, reecting the d 1 -geometry of real Euclidean space, there are many more d 1 -geodesics than just the unique psh geodesic (e.g., take the reparametrization of the concatenation of the geodesic joining φ 0 and P (φ 0 , φ 1 ), and the geodesic joining P (φ 0 , φ 1 ) and φ 1 ). The fact that our segment is maximal at least ensures that it is maximal in the set of d 1 -geodesics which are also psh segments. Lemma 4.3.2.1. Let φ 0 ≤ φ 0 , φ 1 ≤ φ 1 be continuous psh metrics, and denote by φ t , φ t the maximal psh segments joining them. Then, for all t, φ t ≤ φ t .

Proof. By denition, if φ t is maximal, it bounds from above all segments joining endpoints bounded above by φ 0 , φ 1 , and the result follows.

Therefore, the net k → φ k t is monotonous, where φ k t is the maximal psh segment joining φ k 0 and φ k 1 . We claim that

φ t : t → lim k φ k t is our desired geodesic segment.
Proof of Theorem 4.1.3.1. If no psh segment exists between φ 0 and φ 1 , the rst statement of the Theorem is proven.

Assume now that there exist psh segments between φ 0 and φ 1 . Let t → ψ t be such a segment. It is then a decreasing limit of a net of psh segments t → ψ k t . For all k, let t → ψ k t denote the maximal psh segment joining ψ k 0 and ψ k 1 . By maximality, we have for all t, k,

ψ k t ≤ ψ k t .
In particular, lim k ψ k t ≤ lim k ψ k t , and both are psh segments between φ 0 and φ 1 . This shows that one needs only consider limits of maximal psh segments. Furthermore, by the same argument, one needs only consider sequences with endpoints equal to φ 0 and φ 1 . Therefore, we must show that given any two nets of maximal segments φ k t , ψ k t , such that the endpoints converge to φ 0 and φ 1 , the limits are equal for all t:

lim k φ k t = lim k ψ k t . But lim k φ k t = lim n sup{ϕ k t psh segment between φ k 0 and φ k 1 },
Consider two metrics φ 0 , φ 1 ∈ C 0 ∩ PSH(L). Their relative Monge-Ampère energy is the quantity

E(φ 0 , φ 1 ) = 1 (L d )(d + 1) d i=0 ˆX(φ 0 -φ 1 ) (dd c φ 0 ) i ∧ (dd c φ 1 ) d-i .
Note that we have a cocycle identity

E(φ 0 , φ 1 ) = E(φ 0 , φ ) + E(φ , φ 1 )
for any other continuous psh metric φ . Having xed a continuous psh metric φ ref on the right, E(φ) := E(φ, φ ref ) can be seen as an operator on C 0 ∩ PSH(L) which is also a primitive of the Monge-Ampère operator MA : φ → (dd c φ) d . It admits a (possibly innite) extension to PSH(L) via

E(φ) = lim k→∞ E(φ k ),
where φ k is a net of continuous psh metrics decreasing to φ, which always exists by [START_REF] Blocki | On regularization of plurisubharmonic functions on manifolds[END_REF], [Dem92]. The space of nite-energy metrics is the space

E 1 (L) = {φ ∈ PSH(L), E(φ) is nite}.
By the cocycle identity, this space does not depend on the choice of a reference metric. From the work of Darvas, we know this space to admit (amongst others) a d 1 -type complete metric space structure via d 1 (φ 0 , φ 1 ) = E(φ 0 ) + E(φ 1 ) -2E(P (φ 0 , φ 1 )), where P (φ 0 , φ 1 ) is the envelope P (φ 0 , φ 1 ) = sup {φ ∈ PSH(L), φ ≤ min(φ 0 , φ 1 )).

It will be more practical to use a dierent expression of the Monge-Ampère energy, just as we did in the non-Archimedean case, as a dierence of absolute Deligne pairings.

Relative nite-energy metrics and extended Deligne

pairings.

We consider a holomorphic submersion between complex manifolds π : X → Y • the change of metric formula: given another continuous psh metric φ 0 on L 0 , we have The last formula shows that the metric φ 0 , . . . , φ d X/Y is positive. One also notices that a metric on a trivial Deligne pairing O X , L 1 , . . . , L d X/Y can be identied with a genuine function on the base Y , upon evaluation against the trivial section 1.

φ 0 , . . . , φ d X/Y -φ 0 , . . . , φ d X/Y = π * ((φ 0 -φ 0 )(dd c φ 1 ∧ • • • ∧ dd c φ d ))
Assume for the moment that Y is a point. In that case, Deligne pairings can be seen as complex lines together with a Hermitian norm. In this setting, we will omit the subscript • X/Y . Using the change of metric formula, one can see the relative Monge-Ampère energy between two continuous psh metrics on a xed line bundle L over X as a dierence of Deligne pairings:

(d + 1)E(φ 0 , φ 1 ) = φ d+1 0 -φ d+1 1 ,
which suggests that the Monge-Ampère energy should be seen intrinsically as a genuine (Hermitian) metric

(d + 1)E(φ) = φ d+1
on the line L d+1 .

We now return to arbitrary Y . The change of metric formula suggests that the Deligne pairing construction could possibly make sense in a larger class of metrics, where each φ i has brewise nite energy. This motivates the following denition. Denition 5.1.2.1. Let L be a relatively ample line bundle on X. We dene the class of relative nite-energy metrics

E 1 X/Y (L)
to be the class of plurisubharmonic metrics φ on L such that, for all y ∈ Y , φ y ∈ E 1 (L y ). Here, L y is the restriction of L to the bre π -1 (y).

Since we have required plurisubharmonicity on all of L, it follows that any metric in E 1 X/Y (L) can be approximated by a decreasing net of continuous psh metrics on L. In particular, such metrics admit Deligne pairings.

Theorem 5.1.2.2. Let π : X → Y be a holomorphic submersion between complex manifolds of relative dimension d, and let (L i ) d i=0 be a collection of d + 1 relatively ample line bundles on X. There exists a unique extension of the Deligne pairing construction to metrics in E 1 (L i ) X/Y , which is multilinear, symmetric, stable upon restriction to a smaller open set on the base, and such that the change of metric formula (5.1) holds. Proof. We rst restrict to an open set U on the base Y , so that we may apply Demailly regularization on π -1 (U ). Fix for each i a metric φ i ∈ E 1 X/U (L i ), and let k → φ k i be a sequence of continuous psh metrics on L i decreasing to φ i . We claim that the sequence

k → φ k 0 , . . . , φ k d X/U
decreases to a nite-valued metric on U , independent of the choices of approximating sequences, which denes our construction restricted to U . Assuming This Theorem is proven via families of Bergman kernels. We recall some facts on this topic. For all z on the base, picking a smooth, strictly psh metric φ on L endows the H 0 (kL z +K Xz ) (for all positive integers k) with a Hermitian norm

s z 2 φ,z,k = ˆX |s z | 2 e -kφz ,
for |s z | 2 e -kφz is indeed a measure on X z . We may now pick a basis (s j,z ) which is orthonormal for ζ φ,z,k , and dene the Bergman kernel

B φ,z,k := j |s j,z | 2 .
There are two key points regarding this object. The rst one, which is easier to see, is that B φ,z,k is independent of the choice of such a basis. The second, much deeper point is that as we move on the base, the Fubini-Study metrics

φ k : z → F S k (ζ φ,z,k ) := k -1 log B t,z,k
vary plurisubharmonically, i.e. dene a globally psh metric on L. This is a particular case of [BP08, Theorem 0.1]. We will return to this construction shortly.

We now turn to some facts regarding spaces of norms. Given any two Hermitian norms ζ 0 , ζ 1 acting on a complex nite-dimensional vector space V , it is a well-known fact that one may nd a basis (s j ) j of V which is orthonormal for ζ 0 and also orthogonal for ζ 1 . One may then dene a distinguished segment t → ζ t of Hermitian norms by requiring ζ t to be the unique norm orthogonal in the basis (s j ) j , and such that for all j, s j t = s j t 1 .

Such segments are in fact geodesic for various d p -type metric structures on the space of Hermitian norms on V , but we will not need that fact.

We can consider as before the psh geodesic t → φ t,z joining φ 0,z and φ 1,z in E 1 (L z ). Then, a result of Berndtsson ([Berndt09, Theorem 1.2]) states that the Fubini-Study metrics from before approximate the geodesic φ t uniformly in t: there exists a constant c = c(z) such that

|F S k (ζ t,k,z ) -φ t,z | ≤ c(z) • k -1 log k.
(5.3)

One would like to have such an approximation to be (Y -locally) independent of the variable on the base. Firstly, on reading the proof of [Berndt09, Theorem 1.2], one notices that the constant c depends only on the endpoints φ 0,z and φ 1,z , so that our problem reduces to knowing whether one can nd c such that, for all z in some compact U in Y ,

|F S k (ζ 0,k,z ) -φ 0,z | ≤ c • k -1 log k
(and similarly for t = 1; the proof is symmetric.) This follows from adapting the general uniform Bergman kernel asymptotics result [MMBook, Theorem 4.1.1] to the case of varying complex structure, along the same lines as explained in the proof of [START_REF] Ma | Superconnection and family Bergman kernels[END_REF]Theorem 1.6]. The proof of Theorem 5.1.3.1 is now a matter of adequately piecing together all the previous results.

Proof of Theorem 5.1.3.1. Let φ 0 , φ 1 ∈ E 1 X/Y (L), which we assume to be smooth and strictly psh (while the general case follows from regularization), and consider the families of brewise psh geodesics t → φ t . As explained before, for any compact U on the base, and all z ∈ U , there exists a c independent of z such that

|F S k (ζ t,k,z ) -φ t,z | ≤ c • k -1 log k,
by [Berndt09, Theorem 1.2] and the uniform Bergman kernel asymptotics. Furthermore, as also discussed, the families

(t, z) → F S k (ζ t,k,z )
have plurisubharmonic variation in z and t. Combined with the above estimates, this means that, over π -1 (U ) × A (where π : X → Y is the structure morphism of X over Y , and A is the annulus corresponding to [0, 1]), the segment t → φ t seen as a metric on L × A can be uniformly approximated by continuous psh metrics on L × A, furthermore S 1 -invariant under the second variable. This settles the rst statement of the Theorem.

That φ t would be the unique segment such that

dd c t (π y ) * Φ d+1 X×A/Y ×A = 0
for all y ∈ Y follows by denition of the brewise psh geodesic segments (they are themselves characterized as the unique segments in each E 1 (X y , L y ) along which the Monge-Ampère energy is ane). for all z in π -1 (U ).

Remark 5.1.4.2. One sees from this denition that a decreasing limit of relatively maximal psh metric is also relatively maximal.

Remark 5.1.4.3. Let M be a compact Kähler manifold together with an ample line bundle L M . Let [0, ∞) t → φ t be a psh ray of psh metrics on L M . Seen as a S 1 -invariant psh metric on the product L × D * , φ is relatively maximal in our sense if and only if it is "geodesic" in the sense of [BBJ].

A nice way to generate relatively maximal metrics is via Perron-Bremmermann envelopes, as we now prove. We extend our setting slightly, to allow for singular bres, which will be useful later on. We state our result in maximal generality, but the case to keep in mind is that of a holomorphic submersion over the punctured disc with a singular bre over zero.

Theorem 5.1.4.4. Let π : X → Y be a holomorphic projective surjective morphism. Let Ω be a relatively compact, smooth open subset of Y , such that π is a submersion above (hence near) ∂Ω. Let L be a π-ample line bundle on X. Let φ be a continuous collection of brewise psh metrics on π -1 (∂Ω).

We then have that: 1. if there exists a continuous psh extension of φ to all of π -1 (Ω), then there exists a (unique) relatively maximal continuous psh extension of φ to all of π -1 (Ω);

2. if Ω is dened as {ρ < 0}, where ρ is a smooth strictly psh function on Y , such that ∇ρ = 0 whenever ρ = 0, then a continuous psh extension as above exists. Proof of Theorem 5.1.4.4. The hypotheses in the Theorem give that π -1 ( Ω)

is a manifold with boundary, which we denote M := π -1 ( Ω), and whose boundary is π -1 (∂Ω), which we denote ∂M := π -1 ( Ω). We will nally write M := π -1 ( Ω).

1. Existence of a continuous relatively maximal metric, assuming existence of a subsolution.

We claim that the envelope

Pφ = sup * {ψ ∈ C 0 ∩ PSH(L| M ), ψ ≤ φ
f 0 = φ -(φ ref )| ∂M .
Since dd g ≥ -ω, the Laplacian ∆ ω g of g with respect to ω is bounded below by -d -1. Let f be the (continuous) solution on M to the Dirichlet problem

∆ ω f + (d + 1) = 0, f | ∂M = f 0 .
We then have that ∆ ω (g -f ) ≥ 0, which implies by the maximum principle that sup

M (g -f ) = sup ∂M (g -f ),
while this supremum is nonpositive since ψ = g + φ ref is a candidate for the envelope Pφ. We then have that g ≤ f on all of M , and this is true for any candidate ψ, so that Pφ ≤ φ ref + f on X.

We now look at continuity on M . Let φ denote a continuous psh extension of φ to L| M . We x a very small ε > 0, and dene

U = {Pφ * < φ + ε},
which is the complement of a small compact set containing ∂Ω. By regularization (e.g. [BouL2, Theorem 3.8]), we can nd a sequence ψ k ∈ C 0 ∩PSH(L| U ) which decreases to Pφ. Now, by Dini, using compactness, we have that U is covered by nitely many of the U k = {ψ k < φ + ε} (since such inequality holds, for all z ∈ M , and for all large enough k z ).

In particular, for large enough k, one has ψ k < φ + ε. We now dene ψk := max(ψ k -ε, φ), which is dened on all of M . For all k, ψk is continuous, as ψ k is continuous away from the boundary and φ is continuous everywhere (in particular, near and up to the boundary). Furthermore, ψk is equal to φ on the boundary, so that

ψ k -ε ≤ ψk ≤ Pφ ≤ Pφ * ≤ ψ k .
f (z) + a • log |z| -g(0) -log |z| ≤ g(s) -g(0) s ≤ lim t→∞ g(t) t = f -a.
Adding a then concludes the proof.

Plurisubharmonic metrics on degenerations.

Fix now a degeneration π : X → D * , endowed with a relatively ample line bundle L. We will take our interest to plurisubharmonic metrics on L, and in particular their singularities. However, a general psh metric on a degeneration can behave very poorly near the singularity, even though we have assumed existence of an analytic model of X. Thus, we need to enforce a rather natural growth condition on such psh metrics, akin to that of linear growth for geodesic rays.

Denition 5.2.3.1. We say that a psh metric φ on L has logarithmic growth if there exists a model (X , L) of (X, L) such that φ extends as a psh metric on L.

We will write PSH(L) for the space of psh metrics of logarithmic growth on L. If it comes to be necessary, we will rather write PSH(X, L) when considering the space of (non-necessarily of logarithmic growth) psh metrics on L. We will soon show that PSH(L) has many desirable properties. We will also shortly explain our terminology. We begin with the following result:

Lemma 5.2.3.2. Given a psh metric φ on L, the following are equivalent:

(i) φ has logarithmic growth, i.e. there exists a model (X , L) such that φ extends to a psh metric on L;

(ii) for all models (X , L) of (X, L), there exists a constant c = c(X , L) such that φ + c • log |z| extends to a psh metric on L;

(iii) there exists a model (X , L) and a smooth metric φ ref on L such that

ρ * φ(z) ≤ φ ref (z) + O(log |z|)
as z → 0, where ρ denotes the isomorphism between X and X -X 0 ;

(iv) for all models (X , L) of (X, L) and all smooth metrics φ ref on L, (

Proof. By classical results of pluripotential theory, (i)⇔(iii) and (ii)⇔(iv).

Since (iv)⇒(iii) is immediate, we only need to prove (iii)⇒(iv). Assume that

ρ * φ(z) ≤ φ L (z) + O(log |z|)
for a smooth reference metric φ L on L. Pick another model (Y, M) together with a smooth metric φ M . Note that the equation above holds if and only if the same equation holds for the pullbacks of φ L and ρ * φ to a higher model. Thus, we pick a model (Z, N ) dominating both via π X : Z → X , π Y : Z → Y.

There exists a unique Cartier divisor D supported on the special bre Z 0 such that π * X L + D = π * Y M, and given a local equation f D for D, we have

π * X φ L ≤ π * Y φ M -log |f D | + O(1) ≤ π * Y φ M + O(log |z|). Thus, π * X ρ * φ ≤ π * X φ L + O(log |z|) ≤ π * Y φ M + O(log |z|),
as desired.

Remark 5.2.3.3. The above result shows that one could equivalently dene our growth condition using some xed reference data (X ref , L ref ), using e.g. point (ii). In the isotrivial case, there furthermore exists some very natural reference data: the "trivial model" given by the product family of the generic bre with the whole disc.

Example 5.2.3.4. Let [0, ∞) t → φ t be a ray of psh metrics on an ample line bundle L over a xed variety X. It may be identied as a psh metric Φ over the trivial model (X ×D * , L×D * ), by setting Φ z = φ -log |z| . In this case, the logarithmic growth condition is merely the usual linear growth condition on psh rays.

We then have as an immediate Corollary:

Corollary 5.2.3.5. The space PSH(L) is stable under limits of decreasing nets, nite maxima, and addition of constants. It is furthermore the smallest such set containing all psh metrics on L which admit a locally bounded extension to some model (X , L) of (X, L).

Proof. All of those properties are seen to preserve characterization (iv) above, having xed some reference model. To show that it is the smallest set closed under those operations, only the statement about decreasing nets could a priori be delicate. Given a metric φ ∈ PSH(L), (i) shows that it extends as a genuine metric on some model (X , L), and Demailly's regularization Theorem yields a decreasing sequence of smooth (in particular locally bounded) psh metrics decreasing to the extension of φ, which shows in particular that φ belongs to the closure of the set of locally bounded psh metrics on L, proving our result.

5.2.4

The main setting, and some important examples.

We begin with some notation. Let π : X → D * be a degeneration together with a relatively ample line bundle L. We now, and for the remainder of this article, x some reference boundary data φ ∂ , which is the restriction to the boundary π -1 (S 1 ) of a smooth psh metric on L. This is a minor distinction which will allow us to later obtain a genuine metric structure on a particular subspace of PSH(L), rather than a pseudometric structure, and therefore we will assume that a metric in PSH(L) has boundary data equal to φ ∂ . We will dene

E 1 (L) = E 1 X/D * (L) ∩ PSH(L)
to be the space of brewise nite-energy metrics in PSH(L). We also set

Ê1 (L) = {φ ∈ E 1 (L),
Ê1 (L) → Ê1 ( L d+1 X/D * ).
Let a 01 be such that d 1 (φ 0,z , φ 1,z ) + a 01 log |z| is bounded above on the punctured disc, and dene similarly a 02 , a 21 . We have by the triangle inequality of the brewise metric d 1

d 1 (φ 0,z , φ 1,z ) ≤ d 1 (φ 0,z , φ 2,z ) + d 1 (φ 2,z , φ 1,z )
for all z in D * , and in particular d1 (φ 0 , φ 1 ) ≤ d1 (φ 0 , φ 2 ) + d1 (φ 2 , φ 1 ), as desired. Finally, assuming d1 (φ 0 , φ 1 ) = 0, Lemma 5.2.5.3 shows that we must have φ 0 = φ 1 .

d 1 (φ 0,z , φ 1,z )+(a 02 +a 21 ) log |z| ≤ d 1 (φ 0,z , φ 2,z )+d 1 (φ 2,z ,

Completeness.

We now prove completeness of our space.

Theorem 5.2.6.1. The metric space ( Ê1 (L), d1 ) is complete.

In order to prove this, discuss possible topologies for E 1 (L).

Remark 5.2.6.2 (Topologies on E 1 (L)). We have already considered the topology of brewise d 1 -convergence on E 1 (L). There is a yet ner topology, that of locally uniform brewise d 1 -convergence, by which φ k converges to φ if, for all relatively compact open sets U in X, d 1 (φ k,z , φ z ) → 0 uniformly in z on U . In between the two, there is the topology of "base-locally" uniform brewise d 1 -convergence, which is the same but over the π -1 (U ) with U relatively compact open in D * . By the previous Lemma, the latter is equivalent to the topology induced by d1 on Ê1 (L)! and U r respectively. By Proposition 5.1.4.7, those functions are harmonic, and for all s ∈ (0, 1), the functions u r , r > s increase over U s to u, which implies the convergence to be uniform (as an increasing sequence of harmonic functions over a compact set). Now, by harmonicity, for r > s, the integrals whose slopes we have seen to coincide, proving our statement that d 1 ( φz , φ z ) = O(log |z|). Therefore, φ is our desired envelope P (φ). Finally, if φ is not continuous, we extend it to some model (X , L), and a decreasing approximation by continuous metrics φ i on L gives a sequence of relatively maximal metrics φi decreasing to some relatively maximal metric φ which has the desired properties, as we show now: dene u i , u i r and v i as above for φ i , and u, u r and v for φ. By monotonicity of Deligne pairings along decreasing nets, we have that u i → u, u i r → u r and v i → v decreasingly; we then have for all r > s ∈ (0, 1) and all positive integers i that 5.3 The non-Archimedean limit.

We move away from relatively maximal and nite-energy metrics for the moment, and focus on the space PSH(L). The purpose of this Section is to show that there is a natural map from this space to a certain space of non-Archimedean metrics.

Degenerations as varieties over discretely valued elds.

Dating back to ideas of Berkovich ([Berk94], [START_REF] Vladimir | A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures[END_REF]), objects such as degenerations and analytic models thereof can be interpreted as varieties over the eld C((t)) (see also [Fav], [START_REF] Boucksom | Tropical and non-Archimedean limits of degenerating families of volume forms[END_REF]). For clarity, we will from now on write K = C((t)) and R = C[[t]].

Pick a degeneration π : X → D * and an analytic model π : X → D of X. As

X is projective, it can be embedded in some P n × D, where it is presented by a nite number of homogeneous polynomials with coecients in the set of holomorphic functions on D * that are meromorphic at zero. Since this set of functions can be identied with the eld K of complex Laurent series, one can then view X as a variety X K over the eld K. Similarly, X can be presented by nitely many homogeneous polynomials with coecients in O(D), i.e. holomorphic functions over the disc, so that it can be identied with a variety X R over R.

Fixing a psh model metric φ L on L an K , one can identify psh metrics on L an K with "L-psh" functions on X an K , via φ ↔ φ -φ L . We dene more generally the set of L-psh functions to be the reunions of all L-psh functions for all nef models L of L.

Any vertical ideal sheaf a on a model X of X denes a function log |a| on X, [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF]). Let (X , L) be a model of (X, L). Let a be a vertical ideal sheaf on X , such that L ⊗ a is globally generated. Then, φ L + log |a| is a psh metric on L an K .

The main result.

We are now equipped to describe the main construction of this Section. We x a metric φ ∈ PSH(L). Given any divisorial point ν E associated to the component E of a model X of X, we know that φ+a log |z| extends to a metric over E for some a ∈ R. Pick a psh metric φ E with divisorial singularities of type E i on X , i.e. locally of the form

φ E = log |f E | + O(1),
where f E is a local equation for E. We can then dene a generic (signed) Lelong number ϕ NA (ν E ) = ord E (φ) := -sup{c ≥ 0, φ + a log |z| ≤ c • φ E + O(1) near E} + a.

(5.5) By linearity, this is independent of the choice of such an a. Performing this construction over all possible E captures the singularities of φ along all possible models of X . Our main result for this Section is then the following: Theorem 5.3.3.1. Let X be a degeneration together with a relatively ample line bundle L. The Lelong numbers of a metric φ ∈ PSH(L) dene a function on X div , which admits a unique L-psh extension, giving a map which is furthermore lower semicontinuous and order-preserving.

Some preliminaries.

We now prove some auxiliary results that will be useful in the proof of Theorem 5.3.3.1. We rst show that multiplier ideals of psh metrics on L give L an K -psh functions.

Lemma 5.3.4.1. Let φ be a metric in E 1 (L). Let (X , L) be a model of (X, L) such that φ extends as a psh metric on L. Then, up to restricting to a slightly smaller disc, for all m, the multiplier ideal

a m = J (mφ)
is vertical, and there exists an integer m 0 (depending only on L and not on m or φ) such that (m + m 0 )L ⊗ J (φ) is globally generated on X .

Proof. Since φ in particular has brewise nite energy, it has zero Lelong numbers on all bres. As a consequence, φ has zero Lelong numbers on all of X -X 0 , as Lelong numbers cannot increase upon evaluating them on a larger space. Skoda's integrability theorem ([Sko, Theorem 1], see also (which is proven exactly as in the trivially valued case), one has

m • ϕ NA (ν E ) ≤ u m (ν E ) ≤ m • ϕ NA (ν E ) + A X (ν E ),
where A X is the log discrepancy function as before. The sequence ψ m is therefore a sequence of L an R -psh functions converging pointwise on X div to ϕ NA . To show that ϕ NA is L an R -psh, it is then enough to prove that we can have this sequence be decreasing. By subadditivity of multiplier ideals we have J (2mφ) ⊆ J (mφ) 2 , thus ψ 2m ≤ 2ψ m , and as φ m ≤ 0,

ψ 2m ≤ 2(m + m 0 ) 2m + m 0 ψ m ≤ ψ m .
Picking the subsequence i → ψ 2 i therefore yields a decreasing subsequence converging to ϕ NA , as desired. We then set φ NA := ϕ NA + φ L , which concludes our proof.

5.3.6 Locally bounded metrics in the non-Archimedean limit.

We now begin studying the behaviour under the map (•) NA of the class of metrics φ, such that there exists a model (X , L) of (X, L) on which φ admits a locally bounded extension.

Proposition 5.3.6.1. Let φ ∈ PSH(L). Then, 1. φ extends to a psh metric on a model (Y, M) of (X, L) if and only if φ NA ≤ φ M ; 2. φ extends to a locally bounded psh metric on (Y, M) if and only if φ NA = φ M .

Proof. Note that it is equivalent to show the following: given (X , L) an analytic model of (X, L) and ψ be a reference metric admitting a locally bounded extension to L, (1) holds if and only if φ NA -ψ NA ≤ φ M -φ L , and

(2) if and only if we have equality. This will allow us to work at the level of functions and relatively to another model, which is easier.

Assume rst φ to extend to a psh metric on M. φ NA (ν) -ψ NA (ν) ≤ φ M (x) -φ L (x).

In the case where φ admits a locally bounded extension, then there is also a lower bound, which shows by the same argument that φ NA = φ M -φ L . The converse is obtained by uniqueness of the Siu decomposition of φ on X .

5.4 Finite-energy spaces and the Monge-Ampère extension property.

5.4.1 The Monge-Ampère energy in the non-Archimedean limit.

In the trivially-valued setting, we have already seen that a metric in E 1 (L) coincides with a nite-energy psh geodesic ray t → φ t . Two natural "asymptotic" energies arise:

(recall how we dened the model metric φ L in Section 5.3.2). But φ is locally bounded near the central bre of L, so that u is locally bounded near zero, which implies û = 0 as desired.

Remark 5.4.1.3. We will occasionally refer to a metric satisfying the statement of Theorem 5.4.1.2 as satisfying the Monge-Ampère extension property. We also remark that the proof of the Theorem works more generally for arbitrary Deligne pairings(!): given d + 1 pairs of relatively ample line bundles L i on X and metrics φ i ∈ E 1 (L i ) admitting locally bounded extensions to some model of L i , one has ( φ 0 , . . . , φ d X/D * ) NA = φ NA 0 , . . . , φ NA d .

The fact that the slopes are well-dened follows as in the proof of the above Theorem from the general property (5.2) of Deligne pairings! In Section 5.4.4, we will show how to extend this result to the class of metrics satisfying the Monge-Ampère extension property.

5.4.2 Hybrid maximal metrics: existence and uniqueness.

We now study hybrid maximal metrics. Such metrics can be described as being relatively maximal, but with boundary values prescribed both at the complex boundary of X and at the "asymptotic" or non-Archimedean boundary. We will then see that they correspond exactly to metrics satisfying the Monge-Ampère extension property.

Denition 5.4.2.1. Let φ ∈ Ê1 (L). We say that φ is hybrid maximal if for any ψ ∈ E 1 (L) such that ψ NA ≤ φ NA and lim sup(ψ -φ) ≤ 0 near the boundary of X, we have ψ ≤ φ.

Remark 5.4.2.2. We show how to relate our terminology with that of [BBJ],

which deals with special cases of our objects:

• a geodesic ray in [BBJ] is a relatively maximal C * -invariant (logarithmic growth) psh metric on a line bundle over a test conguration in our article;

• a maximal geodesic ray in [BBJ] is a hybrid maximal C * -invariant (logarithmic growth) psh metric on a line bundle over a test conguration in our article.

The "hybrid" refers to (e.g.) the work of Boucksom-Jonsson, in which a hybrid property is a property that passes well from the complex setting to the non-Archimedean limit. Other possible denominations could be "Lelongmaximal" or "maximal in the non-Archimedean limit", but both of those seem to focus more on the limit behaviour while we require our metric to also be maximal in the complex world.

Theorem 5.4.2.3. For any Φ NA ∈ E 1 (L an ), there exists a unique metric φ ∈ Ê1 (L) such that φ NA = Φ NA . Proof. The proof is in two parts. We begin with the assumption that the non-Archimedean metric is a model metric, and construct the unique solution via adapted envelope techniques (inspired by [Berm16, Proposition 2.7]). Then, for the general case, we use properties of the Monge-Ampère energy.

First step: the model case. Assume thus Φ NA to be a model metric corresponding to a model (X , L) of (X, L). Denote by φ the "Perron-Bremmermann-Lelong" envelope dened as the supremum of all metrics ψ ∈ E 1 (L) with lim z→ξ ψ(z) ≤ φ z for all ξ ∈ ∂X, and ψ NA ≤ φ NA .

We begin with a claim that φ so dened belongs to E 1 (L). Note that if we can show that it is plurisubharmonic, then it necessarily has logarithmic growth, as the supremum of a family of metrics with logarithmic growth, and it is by denition relatively maximal. Furthermore, the brewise niteenergy condition will also immediately follow, so that we need to focus on the plurisubharmonicity. Let ψ be in the class of contributions to the supremum above. The hypothesis that ψ NA ≤ Φ NA implies via Proposition 5.3.6.1 that ψ extends with at worst analytic singularities as a psh metric on L. We therefore see φ to be the restriction of a metric φ X dened as the supremum of all metrics on L, with the same boundary conditions as above on ∂X , and extending with at worst analytic singularities over the central bre of X .

That the envelope satises our claim is then a particular case of Theorem 5.1.4.4 (which allows singular bres!).

Finally, the second case of Proposition 5.3.6.1 together with the non-Archimedean maximality assumption ensure that it is hybrid maximal, provided we can show that for any model metric Φ NA there exists a metric ψ NA ∈ E 1 (L) with ψ NA = Φ NA . But this also follows from the same Lemma, since one only has to choose ψ to be a psh metric with a locally bounded extension to L. That φ is the unique hybrid metric given our data follows again from the extremal characterization.

Second step: the general case. The general case again proceeds by approximation: we pick a sequence of model metrics φ NA i decreasing to Φ NA , and their associated hybrid maximal metrics φ i in Ê1 (L), which exist and are unique by the rst part of the proof. The φ i then give a decreasing sequence of metrics by maximality. We write φ for their limit and φ NA the non-Archimedean metric it denes. Since the mappings φ → φ NA are order-preserving, we nd φ NA ≤ φ NA k for all k, i.e.

φ NA ≤ Φ NA .

(5.6) Fix a model (X , L) of (X, L), so that φ and the φ k extend to L (with singularities). Fix a trivialization τ of L d+1 which by Corollary 5.2.4.4 shows that φ is a relatively maximal metric. Furthermore, we know that PSH(L) is closed under decreasing limits: φ thus has logarithmic growth. To establish existence, i.e. to show that φ is our desired solution, we now only have to show that φ NA = Φ NA . Using [Reb20b, Proposition 6.3.2], this is proven provided we can show that E NA (φ NA ) = E NA (Φ NA )

(5.9) by (5.6). One inequality is immediate from the same equation (5.6) and monotonicity of E NA :

E NA (φ NA ) ≤ E NA (Φ NA ).

From (5.8) we have Ê(φ) = E NA (Φ NA ) -E NA (φ L ),

(5.10) so that we have the other inequality (hence (5.9)), provided we can show that Ê(φ) ≤ E NA (φ NA ) -E NA (φ L ). Example 5.4.6.5. This allows us to obtain convexity of the (trivially-valued) non-Archimedean K-energy modulo the entropy approximation conjecture, as follows. Pick a compact Kähler manifold X 1 together with an ample line bundle L 1 , and consider the trivially-valued analytication (X an 1 , L an 1 ) as before. One then introduces the non-Archimedean entropy as

H NA (φ NA ) = ˆX A X MA(φ NA ),
where A X is the log-discrepancy function on X an identied with a space of semivaluations ([BJ18a]), and φ ∈ E 1 0 (L an 1 ). The entropy approximation conjecture states that, given φ NA ∈ E 1 0 (L an 1 ), there exists a sequence φ NA k of model metrics converging to φ NA such that H NA (φ NA k ) → H NA (φ).

Now, Chi Li ([Li, Conjecture 1.6]) shows that, assuming this conjecture, the entropy H NA is exactly the non-Archimedean (radial) extension of the usual complex entropy functional. Adding the energy part that makes up the Mabuchi K-energy, which is convex along complex geodesics, and using Example 5.4.4.2, our previous result shows that the non-Archimedean Kenergy is convex along non-Archimedean geodesics if the conjecture holds.

As things currently stand, extension of the K-energy is only known for rays that generate model metrics, via [BHJ16, Theorem 3.6]. However, the non-Archimedean geodesics of [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF] do not remain in the space of model metrics even if the endpoints are, much as geodesics between Kähler potentials are merely C 1, 1.

Example 5.4.6.6. In [BLXZ], Blum-Liu-Xu-Zhuang prove, using algebraic techniques, convexity of the non-Archimedean Ding energy (and other functionals) along geodesics between test congurations ([BLXZ, Theorem 3.7]).

Our heuristic allows us to also recover this result: convexity of the complex Ding energy is a result of Berndtsson ([Berndt15]) while the non-Archimedean extension of the Ding energy follows from [START_REF] Boucksom | Uniform K-stability and asymptotics of energy functionals in Kähler geometry[END_REF].

Remark 5.4.6.7. As a Corollary, existence and uniqueness of minimizers for non-Archimedean energy functionals can also be detected strict convexity results in the complex setting, but some form of uniform strict convexity (in |z|) is required to ensure strict convexity of the limit.

of Kähler-Einstein metrics on a degeneration of Calabi-Yau manifolds does not necessarily vary plurisubharmonically ([CGP19, Theorem 3.1]). One can however take the plurisubharmonic envelope P (φ) of φ, and then the hybrid maximal metric Φ with Φ NA = P (φ) NA . In [START_REF] Boucksom | Tropical and non-Archimedean limits of degenerating families of volume forms[END_REF], Boucksom-Jonsson show that the family of measures MA(φ z ) converge in a certain sense to the non-Archimedean Monge-Ampère measure of some metric ψ NA . We therefore formulate the following result, which would connect our hybrid maximal setting with degenerations of Kähler-Einstein metrics on Calabi-Yau manifolds:

Conjecture 5.4.7.1. ψ NA = P (φ) NA = Φ NA .

196

  Berman-Boucksom-Jonsson ([BBJ]), dont nous généralisons certaines lignes au cours de cette thèse, en donnent une formulation nonarchimédienne ([BBJ, Theorem A, Corollary 5.2]) : l'existence d'une métrique Kähler-Einstein sur X est équivalente à la coercivité d'une fonctionnelle (la fonctionnelle de Ding) sur l'espace des métriques psh non-archimédiennes sur l'analytié de K X par rapport à la valeur absolue triviale sur C. Notons que des généralisations de cette conjecture ont par la suite été étudiés, comme l'existence de métriques cscK qui a été récemment réduit par Li ([Li]) à une conjecture purement de pluripotentiel non-archimédien ; et le cas, plus général, des solitons a été étudié de manière similaire par).

1.1. 1

 1 Basic denitions. Denition 1.1.1.1. Let R be any ring. A multiplicative seminorm on R is a function | • | : R → R + satisfying the following properties:

  trum of two norms ζ, ζ ∈ N (V ). Denition 1.2.2.1. The relative spectrum of ζ and ζ is the set (counting multiplicities) Sp(ζ, ζ ) which contains all real numbers of the form

  as follows. Let | • | be a point in Spec A. One associates to it the ideal a |•| := {a ∈ A, |a| = 0}. This is a prime ideal, and therefore denes a scheme point. The mapping ker(| • |) = a |•| is then well-dened, and continuous by denition.

  The implicit claim that metrics in the image of FS m are continuous is a consequence of[BE, Theorem 7.16]. Fubini-Study operators behave well under ground eld extension: Lemma 2.3.2.2 ([BE, Lemma 7.20]). Let K /K be a complete eld extension. Let ζ ∈ N (H 0 (kL)) for some k. Let L K denote the base change of L to K , and ζ K the ground eld extension of ζ to the base change H 0 (kL K ) = H 0 (kL) K . Then, FS k (ζ K ) coincides with the pullback of FS k (ζ) to L an K .

  Denition 2.3.3.8. The graded supnorm operator N • sends a bounded metric φ to the bounded graded norm (N m (φ)) m .

  Theorem 3.1.3.3. Let ζ • , ζ • be bounded graded norms on R(X, L). Then, the sequence of their relative spectral measures σ k (ζ k , ζ k ) converges weakly to a compactly supported measure on R. Proof. By [CM15, T5.2], our desired limit measure exists provided the hypotheses of [CM15, T4.5] are veried, which means the following: (1) ζ • and ζ • are submultiplicative graded norms;

  and nally (3.3) follows. In general, since our norms are bounded graded, they are at linearly bounded distance from a norm of the form ζ •φ , thereby showing that Φ satises 3.3, and proving the Theorem.

  in the previous Theorem by µ(ζ • ), and the measures µ(k) as µ(ζ k ).

(4. 3 )

 3 Since the composition of FS k • N k preserves inequalities while N k and FS k reverse them, and since we have ψ0 = FS k (N k (ψ 0 )) and ψ1 = FS k (N k (ψ 1 ))

4.3. 2

 2 Proof of Theorem 4.1.3.1. Consider now two metrics φ 0 , φ 1 ∈ PSH(L), and pick decreasing nets φ k 0 , φ k 1 in C 0 (L) ∩ PSH(L) converging to φ 0 , φ 1 .

  Elk90, Théorème I.1.1(d)]); • the curvature formula dd c φ 0 , . . . , φ d X/Y = π * (dd c φ 0 ∧ • • • ∧ dd c φ d ) (5.2) (see [Elk90, Théorème I.1.1(d)]).

  Remark 5.1.4.5. An open subset that satises the second point above is sometimes called a hyperconvex open subset. In particular, D and annuli centered at zero are such open sets. The proof of the rst point follows some ideas dating back to the work of Bedford-Taylor ([BT]), see e.g. [BBGZ, Proposition 6.3], [PS].

|z|=r u s

 s (z) dz are ane functions of log r. Writing v = -log |τ | φ d+1 , we then have |z|=r u s (z) dz = log r log s |z|=s v(z) dz + 1 -log r log s • |z|=1 v(z) dz, (recall how we have dened φ s and u s !). Taking the limit s → 0 using the uniform convergence discussed above yields |z|=r u s (z) dz = -(log r) v + |z|=1 v(z) dz, where v denotes the generalized slope of the subharmonic function v. Taking slopes in this equality, one then nds v = û. Now, since φ ≤ φ, we have d 1 (φ z , φz ) = u(z) -v(z),

  |z|=r

  u i s (z) dz = log r log s |z|=s v i (z) dz + 1 -log r log s • |z|=1 v i (z) dz;furthermore, we may normalize all our sequences so that all the functions involved are nonpositive, thereby allowing us to use monotone convergence and nd|z|=r u s (z) dz = log r log s |z|=s v(z) dz + 1 -log r log s • |z|=1 v(z) dz,so that we may proceed using the same argument as before to show that d 1 ( φz , φ z ) = O(log |z|); that φ is the smallest relatively maximal metric bounded below by φ and satisfying this equality follows again by construction, since decreasing limits of relatively maximal metrics over annuli remain relatively maximal.

  via log |a|(x) = max{log |f (x)|}, where the f run over a set of local generators for a. (In particular, any vertical Cartier divisor D on a model denes such a function.) We then have the following crucial result: Lemma 5.3.2.1 ([

(

  •) NA : PSH(L) → PSH(L an K ),

[ Dem12 ,

 Dem12 Lemma 5.6(a)]) then yields local L 1 -integrability of e -φ , which in turn implies local L p -integrability of e -φ for all ∞ > p ≥ 1, and in particular, for all positive integers m, L 1 -integrability of e -mφ . By [Dem12, Lemma 5.6(a)] again, the multiplier ideals satisfya m,x = O X ,xfor all m and for all x outside of the central bre, i.e. a m is cosupported on the central bre. Now, the global generation statement, follows from a relative equivalent of [Dem12, Proposition 6.27]. We can in fact argue just as in [BBJ, Lemma 5.6]: we must prove that there exists m 0 such that the sheaf (m+m 0 )L⊗J (φ) is πglobally generated. By the relative Castelnuovo-Mumford criterion, having picked a relatively very ample line bundle V on X and an m 0 such that m 0 • L -K X -(d + 1)V is relatively ample (after possibly restricting to a smaller disc), it is enough to show that for all j = 1, . . . , d, R j π * (((m + m 0 )L -jV ) ⊗ J (φ)) = 0 on the disc, which follows from Kodaira and Nadel vanishing. model (X , L) of (X, L). Using a version of the estimate[START_REF] Berman | A variational approach to the Yau-Tian-Donaldson conjecture[END_REF] Lemma B.4]

  Let Z dominate both models via π X : Z → X and π Y : Z → Y, and we have π * Y M = π * X L + D for a unique Cartier divisor D supported in the special bre Z 0 . Since φ extends to a psh metric on (Y, M) if and only if it extends to a psh metric on any model dominating (Y, M), we may without loss of generality focus on Z. Picking a local equation f D for the divisor D obtained as above, φ extends to π * Y M if φ -ψ ≤ -log |f D | + C near Z 0 . Taking generic Lelong numbers with respect to the underlying divisor of a divisorial point ν gives ν(φ) -ν(ψ) ≥ -ν(D), i.e.

X

  /D , and set E(φ z ) := -log |τ (z)| φ d+1 z , E(φ k,z ) := -log |τ (z)| φ d+1 k,z .We will also denote as usual by E NA (φ NA ) the metric (φ NA ) d+1 . Now, by Corollary 5.2.4.4, and Theorem 5.4.1.2, we have for all kE(φ k,z ) = c k • log |z| + H(z), (5.7)where H = H(φ ∂ ) is some function bounded near zero and independent of k. In fact, one can see thatc k = -( (φ NA k ) d+1 -φ d+1 L ) = -(E NA (φ NA k ) -E NA (φ L )).Since the (Archimedean and non-Archimedean) Monge-Ampère energies are continuous along decreasing nets, we haveE NA (φ NA k ) → E NA (φ NA ) while E(φ k,z ) → E( φz )for all z. Combining those with (5.7), one ndsE(φ z ) = -(E NA (φ NA k ) -E NA (φ L )) • log |z| + H(z),(5.8)

(5. 11 )

 11 This inequality follows from a similar argument. Let ψ NA k be a decreasing sequence of model metrics approximating φ NA . Let ψ k denote their associated hybrid maximal metric, and dene E(ψ k ) as before. Now, since for all k φ NA ≤ ψ NA k , by maximality, we have φ ≤ ψ k (5.12) whence E(φ z ) ≤ E(ψ k,z ).

  ) ≤ Ê(ψ k ).

  4.1.2 and the arguments above, Ê(ψ k ) = E NA (ψ NA k )-E NA (φ L )which again upon taking the decreasing limit in the right-hand side (alongwhich E NA is continuous) establishes Ê(φ) ≤ lim k E NA (ψ NA k )) -E NA (φ L ) = E NA (φ NA ) -E NA (φ L )(5.15)

  Norm geodesics. . . . . . . . . . . . . . . . . . . . . . . 1.3 Spaces of norms on graded algebras over non-Archimedean elds. 1.3.1 Bounded graded norms. . . . . . . . . . . . . . . . . . 1.3.2 Asymptotic spectral measures, volumes, and metric structures on N • (V • ) . . . . . . . . . . . . . . . . . . . Completeness of N • with respect to the d ∞ distance. . . Equidistribution of Okounkov points of superadditive functions associated to bounded graded norms. . . . . 3.1.5 Approximation of graded norms generated in degree one via graded norms coming from models. . . . . . . . 3.2 The range of the asymptotic Fubini-Study operator. . . . . . . 3.2.1 Relating asymptotic volumes and Monge-Ampère energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The asymptotic Fubini-Study operator descends to a bijection. . . . . . . . . . . . . . . . . . . . . . . . . . .

	1.3.3 Geodesics between bounded graded norms. . . . . . . .
	1.3.4

1.1 Non-Archimedean elds. . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic denitions. . . . . . . . . . . . . . . . . . . . . . 1.1.2 Classication of complete valued elds. . . . . . . . . . 1.1.3 Maximal completeness. . . . . . . . . . . . . . . . . . . 1.2 Spaces of norms on vector spaces over non-Archimedean elds. 1.2.1 Spaces of norms. . . . . . . . . . . . . . . . . . . . . . 1.2.2 Relative spectra. . . . . . . . . . . . . . . . . . . . . . 1.2.3 Spectral measures, volumes, and metric structures on N (V ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Apartments. . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 3 The range of the asymptotic Fubini-Study operator over general non-Archimedean elds. 3.1 Some preliminary results on approximations of bounded graded norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Bounded graded norms on section rings of semiample Q-line bundles. . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Okounkov bodies associated to section rings. . . . . . . 3.1.3 Okounkov bodies and limit measures. . . . . . . . . . . 3.1.4 4 Geodesics in non-Archimedean pluripotential theory. 4.1 Plurisubharmonic segments in non-Archimedean geometry. . .

  'après les travaux d'Aubin et Yau, l'on sait que si X est canoniquement polarisée (i.e. K X est ample), ou Calabi-Yau (K X est trivial), il existe nécessairement une métrique Kähler-Einstein sur X. Dans le cas canoniquement polarisé, une telle métrique est de surcroît unique.

c'est-à-dire le fait que l'enveloppe psh d'une métrique continue est elle-même continue. C'est une propriété de la paire (X, L), qui est classique dans le cas complexe si X est (par exemple) normale et L est ample ; dans le cas non-archimédien, cette propriété est partiellement conjecturale sous les mêmes hypothèses. Elle est connue, au moins, dans tous les cas pertinents aux applications à la géométrie complexe (c'est-à-dire dans le cas où le corps de base est le corps des complexes muni de la valeur absolue triviale, ou bien le corps des séries formelles à coecients complexes). Nous passons en revue les cas connus dans la Section du Chapitre 2 dédiée aux enveloppes psh non-archimédiennes.

Pour conclure ces préliminaires, décrivons brièvement une application récente de la théorie du pluripotentiel non-archimédien à la géométrie complexe, suivant les travaux de Berman, Boucksom, et Jonsson : une approche variationnelle à la conjecture de Yau-Tian-Donaldson.

L'existence de métriques à courbure constante (en divers sens) est un l rouge de la géométrie diérentielle depuis de nombreuses décennies. Un problème particulièrement intéressant est celui de l'existence de métriques Kähler-Einstein sur une variété Kählérienne compacte X, c'est-à-dire une métrique φ lisse et strictement psh sur le bré canonique K X , dont la (1, 1)-forme ω associée est proportionnelle à sa courbure de Ricci Ric(ω). L'existence d'une métrique Kähler-Einstein est une condition très forte, et nécessite un substrat topologique adéquat : elle nécessite que la classe c 1 (X) soit elle-même signée. DLe cas Fano (K X antiample) est notoirement plus dicile.

Inspiré par diverses métaphores mathématiques (la théorie des quotients GIT, plus particulièrement le théorème de Kempf-Ness et ses conséquences en géométrie symplectique ; ainsi que l'alors récent résultat de Donaldson-Uhlenbeck-Yau, démontrant équivalence entre une notion algébrique de stabilité de brés vectoriels holomorphes et l'existence de connexions hermitiennes de Yang-Mills sur ceux-ci), Donaldson formule alors une conjecture, par la suite ranée et désormais connue sous le nom de conjecture de Yau-Tian-Donaldson. Selon celle-ci, l'existence de métriques Kähler-Einstein sur une Fano X équivaudrait à la positivité de certaines quantités purement algébriques, les invariants de Futaki, de toutes les congurations test de X,

  On traite d'abord le cas de géodésiques entre deux métriques continues, où l'on dénit les géodésiques comme un supremum, et où l'on montre via une transformée de Legendre que celles-ci restent également continues en temps et en espace.On montre ensuite, en adaptant l'approche quantiée de Darvas-Lu-Rubinstein à notre contexte, que l'énergie est ane le long de ces géodésiques, et qu'elles satisfont bien l'équation géodésique. Du côté algébrique, on obtient de manière équivalente des géodésiques au sens fort entre des normes graduées bornées.

où les (s i ) i forment encore une base de sections sans point-base d'une puissance kL, et les λ i , λ i sont des constantes réelles. Ensuite, nous dénissons un segment psh comme une limite décroissante de segments Fubini-Study.

En s'inspirant de la caractérisation extrémale du côté complexe, nous développons dans

[START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF]

(ici le Chapitre 4) une théorie des géodésiques dans les espaces de métriques d'énergie nie non-archimédiennes.

Enn, on étend la construction au cas de deux métriques φ 0 , φ 1 générales d'énergie nie, en les approchant par des suites décroissantes de métriques continues φ k 0 , φ k 1 , et en dénissant la géodésique entre φ 0 et φ 1 comme la limite (nécessairement décroissante, par la caractérisation extrémale mentionnée cidessus) des géodésiques t → φ k t . An de montrer que cette limite existe et est unique, on développe certaines propriétés de l'espace E 1 non-archimédien.

Résumons nos résultats en un énoncé compact :

  5. l'énergie de Monge-Ampère est ane le long du segment φ t , et ce segment est l'unique segment psh joignant φ 0 et φ 1 satisfaisant cette propriété. Si de surcroît les métriques φ 0 et φ 1 sont continues, alors le segment psh maximal les joignant est également continu, en temps et espace. nombreux travaux au cours de la précédente décennie, dus notamment à Berman-Boucksom-Favre-Hisamoto-Jonsson et al. ont été dédiés à l'étude de limites non-archimédiennes de rayons géodésiques de métriques plurisousharmoniques. Plus particulièrement, étant donné une variété projective compacte X munie d'un bré en droites ample L, un rayon géodésique (0, ∞] t → φ t ∈ PSH(X, L) est identié à une métrique psh Φ S 1 -invariante sur le tiré en arrière de L au produit trivial D

	Espaces de métriques sur des dégénérescences de variétés
	complexes.

(E 1 (L), d 1 ) est un espace métrique; 2. il existe un segment psh maximal t → φ t joignant φ 0 et φ 1 ; 3. φ t ∈ E 1 (L) pour tout t ∈ [0, 1]; 4. le segment φ t est une géodésique (au sens métrique) pour la distance d 1 , c'est-à-dire qu'il existe c ≥ 0 satisfaisant d 1 (φ t , φ s ) = c • |t -s| pour tous t, s ∈ [0, 1]; De * × X, où la variable z donnée par la première projection est identiée à notre t via t = -log |z|. La limite en t → ∞ dénit, via l'étude des singularités de Φ, une métrique non-Archimédienne sur l'analytié de X par rapport à la valeur absolue triviale sur C. Cette approche a culminé en la preuve variationnelle de la conjecture de Yau-Tian-Donaldson dans [BBJ]. Le but de mon article [Reb21], ici le Chapitre 5, est de pousser cette étude au cas plus général d'une dégénérescence arbitraire π : X → D * de variétés projectives complexes, pas nécessairement S 1 -invariante ni même isotriviale, munie d'un bré en droits relativement ample L. Il est toujours possible de réaliser X comme une variété algébrique sur le corps C((t)), et un modèle analytique X (i.e. un espace analytique possiblement singulier relatif à D, isomorphe à X en-dehors de sa bre centrale) avec un modèle algébrique, déni sur l'anneau de valuation C[[t]].

  Ê1 hyb (L) est une géodésique psh si et seulement si son image dans E 1 (L an K ) est une géodésique psh non-archimédienne au sens du Chapitre 4. • L'on dispose d'une propriété générale d' "extension de plurifonctionnelles" : étant donné d + 1 brés relativement amples L i sur X, pour tout (d + 1)-uplet de métriques φ i ∈ Ê1 hyb (L i ), on a

	Plan du manuscrit.
	Au Chapitre 1, nous développons les préliminaires purement algébriques
	nécessaires à cette thèse. Nous commençons par des généralités sur les corps
	non-archimédiens. Ensuite, ayant xé un tel corps K, nous étudions les
	propriétés métriques et spectrales des espaces de normes sur les K-espaces
	vectoriels de dimension nie. Enn, nous étendons cette étude aux espaces de
	normes graduées sur des K-algèbres graduées engendrées en degré un. Cette
	dernière partie contient quelques résultats de [Reb20b], ainsi qu'un résultat
	non publié sur la complétude d'un certain espace de normes graduées pour
	une distance de type d ∞ .
	Au Chapitre 2, nous nous penchons sur les aspects géométriques des prélim-
	inaires. Après avoir brièvement rappelé la construction de Berkovich, nous
	( φ 0 , . . . , φ d X/D * ) NA = φ NA 0 , . . . , φ NA d nous penchons sur plusieurs types de métriques sur des analytications de . brés en K-droites. Nous expliquons également les bases de la théorie du
	pluripotentiel non-archimédien.

  K, i.e. a eld together with an absolute value | • |, which is furthermore complete with respect to the topology induced by the latter, and satisfying the ultrametric inequality rening the triangle inequality: for all x, y in K, |x + y| ≤ max(|x|, |y|).

	potential theory); and their perfectoid extensions have recently been at the
	heart of a very rich theory developed by Scholze and others ([Scholze], [FF]
	Fundamental examples of such elds include:
	1. C((t)), the eld of Laurent series with coecients in C, together with
	the t-adic absolute value;
	2. given p prime, Q p , the eld of p-adics, together with the p-adic absolute
	value;
	3. given p prime, F p ((t)), the eld of Laurent series with coecients in
	the nite eld F p , with the t-adic absolute value;
	4. any eld K with the trivial absolute value, equal to 1 on K × , which is
	non-Archimedean.
	Such examples shed some light on the diversity of the non-Archimedean
	world. Uses for such elds are many: p-adic elds are ubiquitous in num-
	ber theory (note the Hasse-Minkowski principle, Hensel's Lemma, Mahler's
	Theorem; as well as applications to p-adic dierential equations and p-adic

  Fields of Puiseux series (i.e. series with exponents bounded below in Q) over C and F p are well-known examples, but they are not Cauchy complete! In the mixed characteristic case, the fundamental example is the eld C

p of complex p-adics; though recently, much attention has been brought to perfectoid elds, i.e. densely valued mixed characteristic non-Archimedean elds with surjective Frobenius on K • mod p. We will give examples closer to our considerations in the next Subsection.

  This denes by [BE, Proposition 1.24(i)] a non-Archimedean norm on V L , which coincides with the original norm ζ on V . Two essential results for us Let ζ be a norm on V , with ground eld extension ζ L on V L . We then have:• if ζ is diagonalizablewith basis (e i ), then ζ L is also diagonalizable with basis (e i ⊗ 1); • the relative spectra of ground eld extensions of norms coincides with with the relative spectra of original norms: for any other norm ζ with ground eld extension ζ L , we have

	are as follows:
	Proposition 1.2.2.4 ([BE, Lemma 1.25, Proposition 2.14(v)]). Let L/K be a eld extension.

Sp(ζ, ζ ) = Sp(ζ L , ζ L ).

The second point follows from the rst, and the fact that the ground eld extension of a norm coincides with the original norm on V . 1.2.3 Spectral measures, volumes, and metric structures on N (V ) Denition 1.2.3.1. The relative spectral measure σ(ζ, ζ ) of ζ and ζ is dened to be the discrete probability measure supported on Sp(ζ, ζ ), that is:

  The distance d 2 also has some importance in the Euclidean picture, which we lightly touch on in the next Subsection.

Remark 1.2.3.3 (Important characterization of the distance d ∞ ). The distance d ∞ (ζ, ζ ) is equivalently characterized as the maximal exponential distorsion between the two norms, or in other words best constant C > 0 such that for all

  If ζ 0 ≤ ζ 0 and ζ 1 ≤ ζ 1 , we then have that for all t ζ t ≤ ζ t .

Proposition 1.2.5.4 (Monotonicity of norm geodesics with respect to endpoints). Let k ∈ N, and set two couples of norms (ζ 0 , ζ 1 ) and (ζ 0 , ζ 1 ) acting on H 0 (kL).

  from Lemma 1.2.5.3 and Lemma 1.2.5.6 we have that the func-

tions t → det ζ t and t → det ζ t are log-convex. Combining this with the equality above, we nd

  , vol(χ t , ζ t ∨ ζ t ) ≤ 0, and we have d 1 (ζ t , ζ t ) = vol(ζ t , ζ t ∨ ζ t ) + vol(ζ t , ζ t ∨ ζ t ) = vol(ζ t , χ t ) + vol(χ t , ζ t ∨ ζ t ) + vol(ζ t , χ t ) + vol(χ t , ζ t ∨ ζ t ) ≤ vol(ζ t , χ t ) + vol(ζ t , χ t ). ζ t , ζ t ≤ χ t , the statement of the corollary holds for the volumes above,

	Since and we have

If the endpoints are comparable in the same order, i.e. ζ 0 ≥ ζ 0 and ζ 1 ≥ ζ 1 ; or ζ 0 ≤ ζ 0 and ζ 1 ≤ ζ 1 , this follows immediately from the previous

Corollary. In the general case, we have to be careful, as the maximum of norm geodesics is not a priori a norm geodesic. However, from Proposition 1.2.5.4, we have that the geodesic

t → χ t joining ζ 0 ∨ ζ 0 and ζ 1 ∨ ζ 1 satises χ t ≥ ζ t ,

ζ t for all t, i.e. χ t ≥ ζ t ∨ ζ t . Therefore

  Later on, we will see how graded norms generated in degree one naturally arise in a geometric context. Denition 1.3.1.3. We will say that a graded norm ζ • on V • is a bounded graded norm if it has at most exponential distorsion with respect to a norm generated in degree one, i.e. there exists a graded norm ζ • generated in degree one on V • , and a constant C > 0 with e -kC ζ m ≤ ζ k ≤ e kC ζ m for all k ∈ N * . We denote by N • (V • ) the set of such graded norms. 1.3.2 Asymptotic spectral measures, volumes, and metric structures on N • (V • )

We now see how to transpose the constructions of Section 1.2.3. Theorem 1.3.2.1. Fix two bounded graded norms ζ •

  We say that two bounded graded norms ζ • and ζ • on V • are asymptotically equivalent, and we write

see e.g.

[START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF] R3.8

]. This justies the following denition.

Denition 1.3.2.4.

  2.4.2.3. A strong motivation to study this class is that mixed Monge-Ampère operators can be extended to E 1 , by the work of Boucksom-

	Favre-Jonsson (see [BFJ15, Section 6.3]).
	As one can see, the relative Monge-Ampère energy shares some properties
	with the relative volume of bounded graded norms. In fact, we have the
	following results:
	Theorem 2.4.2.4 ([BE, Theorem 9.15], [Reb20a, Theorem B]). Let L be a semiample

  • ) and FS(ζ • ) are model metrics, this is essentially proven in[BE]. If ζ • is merely nitely generated, we then show that it can be nicely approximated by graded norms giving rise to model metrics, thereby extending the Theorem to this case. In the general case, inspired by Fujita's lemma, we show that we can approximate the volume of ζ • by the volumes of the ζ

	(r)

  d which is compact, convex, and has nonempty interior. Denition 3.1.2.5. As a consequence of the previous Lemma, one can dene a convex body in R d by projecting the base of the convex cone Cone(Γ(L)) to the last d variables:

  with monomial rst term. Given a bounded graded norm ζ • on L, the individual norm ζ k induces a quotient norm ζ k,α on gr k,α (L). Given any s with a Taylor expansion as above, it is immediate that its class [s] k,α =: s k,α in gr k,α (V • ) contains all elements with such an expansion, and we dene

  Theorem 3.1.4.7. Let L be such that R(X, L) is generated in degree one. Let ζ • , ζ • be two bounded graded norms on L, and for each k ∈ N

* , let ζ (k) • and ζ • (k) denote the bounded graded norms on R(X, kL) generated in degree one by ζ k and ζ k respectively. Then, we have that:

  + C, . . . , s |I| + C) = g(s 1 , . . . , s |I| ) + k -1 C, because those properties are satised by f in the |I| last coordinates, and are preserved upon taking the inmum over the rst coordinate. Composing such a function with our formal tropicalization map naturally yields a psh metric (by Remark 4.2.2.2 below), which proves our result. ) i∈I of H 0 (kL) and a convex PL function f of p = |I| real variables, increasing in each variable, and satisfying f (z 1 + C, . . . , z p + C) = f (z 1 , . . . , z p ) + k -1 C, for all real constants C, then f (log |s 1 |, . . . , log |s p |) is a Fubini-Study metric.

	(In particular, putting aside the
	PL hypothesis, this is precisely the standard proof of the convex inmum
	principle for marginals.)
	Note that our convex PL function g is increasing in each variable and satises,
	for any real constant C,
	g(s 1 Remark 4.2.2.2. We have claimed that, given a basepoint-free basis of
	sections s = (s i

  , φ t psh segment joining φ 0 , φ 1 }, and by Lemma 4.2.3.1, this segment is a psh segment joining φ 0 and φ 1 .

	Proof of Theorem 4.2.1.1 (1)-(2). By Lemma 4.2.3.2, the curve
	t → φ t = sup	tτ + φτ
	τ ∈R
	is equal to
	sup{φ t Therefore, it is a maximal psh segment, and hence is unique. This establishes
	(1). The continuity statement (2) also follows from Lemma 4.2.3.1.
	4.2.4 Quantization with geodesics of bounded graded
	norms.

  They can be joined by the geodesic of graded norms ζ t • , where ζ t k is the norm geodesic joining the N k (φ i ), as in the rst Chapter. By Theorem 1.3.3.1, for all t, ζ t

	•
	is submultiplicative, so that the limit

  The energy is then ane in the diagonalizable case. If the norms are not diagonalizable, we simply note that the geodesics ζ t

	k can be approximated by
	diagonalizable geodesics ζ t k,ε for which

  of relative dimension d. Pick d + 1 pairs (L i , φ i ), where L i is a relatively ample line bundle over X, and φ i is a continuous psh metric on L i . To this data, one associates a line bundle over Y ,

	L 0 , . . . , L d X/Y ,
	together with a metric
	φ 0 , . . . , φ d X/Y
	in a way that is multi-additive, symmetric; the construction furthermore
	commutes with base change (in particular, is stable upon restriction to an
	open set on the base), and satises

  5.1.4 Relatively maximal metrics.Denition 5.1.4.1. Let π : X → Y be a holomorphic submersion with compact Kähler bres. Let L be a relatively ample line bundle on X. We say that a metric φ on L is relatively maximal if it is maximal in the usual sense of Sadullaev (e.g.[KliBook]) on the preimage of any relatively compact open subset of Y . In other words, φ is relatively maximal if and only if, for any relatively compact open subset U of Y , for any relatively compact open subset V of π -1 (U ), and for any psh metric ψ on the restriction of L to π -1 (U ) such that lim sup ψ(z) -φ(z) ≤ 0

as z approaches the boundary of π -1 (U ), then ψ(z) ≤ φ(z)

  on L| ∂M } is our desired relatively maximal, continuous metric on L| M which coincides with φ on L| ∂M . By denition, Pφ is relatively maximal; furthermore, since there exists a continuous subsolution, i.e. a candidate ψ to the envelope which coincides with φ on L| ∂M , Pφ also has the correct boundary values.

We are therefore left to show continuity.

We begin with a continuity estimate near the boundary. Having xed a reference smooth, strictly psh metric φ ref on L, and setting ω = dd c φ ref , we can see any candidate ψ for the envelope Pφ as a continuous ω-psh function g = ψ -φ ref . Fix such a g, and set

  This implies that ψ k converges uniformly to Pφ, i.e. Pφ is continuous on M . Pφ ≤ φ ref + f , as at the end of the rst point of the proof; 2. φ ref + f converges continuously to φ near the boundary, and Pφ is continuous on M ;3. there exists a psh extension φ, ensuring that Pφ = φ on the boundary,In particular, a subharmonic function f with logarithmic growth can be extended as a subharmonic function over the entire disc. In this case, one can dene its generalized (in the sense that it is possibly signed) Lelong number, as follows. Pick a number a as above. The function

	The following estimate will be very useful later on.
	Lemma 5.2.2.5. Let f a subharmonic function f with logarithmic growth on D * . Then, for all z, we have
	Furthermore, since: for t ∈ [0, ∞), is then a convex function of t, and the rate of change |z|=e -t (f (z) + a log |z|), g(t) -g(0) t lim t→∞ f (t) t . Remark 5.2.2.4. As a consequence of Harnack's inequality, f may equiva-lently be computed using the integrals f (z) ≤ log(1/|z|) • f + c, Proof. Dene the function g(t) := sup |z|=e -t (f (z) + a log |z|) as before. Since the rate of change g(t) -g(0) 1. g(t) := sup t
	f (z) dz
	|z|=r
	in place of the suprema.

is thus an increasing nonnegative function of t, which has a nite value as t → ∞. This corresponds to saying that the limit lim r→0 sup |z|=r f (z) + a log |z| -log r exists and is nite.

Denition 5.2.2.2. Given a subharmonic function f with logarithmic growth on D * , we dene its generalized slope (or generalized Lelong number at zero) to be the value

f := lim r→0 sup |z|=r f (z) + a log |z| -log r + a,

where a is a real such that f + a log |z| is bounded near zero. In particular, f is independent of the choice of such an a.

Example 5.2.2.3. In the case of an S 1 -invariant subharmonic function f , i.e. a convex function on [0, ∞), this simply computes the slope at innity where c = sup |z|=1 f (z).

is an increasing function of t, we have for all s ∈ [0, ∞), all |z| = e -s and since, having xed our boundary data, v(0) = 0, we have

  φ is relatively maximal}. For a S 1 -invariant metric φ on L M × D * , seen as a ray [0, ∞) t → φ t , growth has to be of the form c•log |z|+H(z), where H(z) is the solution of the generalized Dirichlet problem over the whole disc with the given boundary data. In particular, it is an ane space isomorphic to R! This agrees with the radial case, where E 1 (L) is simply the set of ane functions on [0, ∞) emanating from the same point, which is isomorphic to the set of possible slopes. This decomposition (in particular, c and H) depends on τ ; but the fact that φ can be decomposed in any trivialization in

	plus nite energy no longer implies logarithmic growth, since there exist har-
	monic functions on the punctured disc that do not have logarithmic growth
	at zero (e.g. the real part of z → e 1/z )!
	Example 5.2.4.1. Although those are seemingly restrictive conditions, they Assuming logarithmic growth, we then have a full description of Ê1 (L) in
	are in fact general enough to encompass the study of maximal geodesic rays. Let (X, L) be a product family (M × D relative dimension zero, since we then see that any (nite-valued) harmonic 1. being in PSH(L) corresponds to the usual linear growth condition; 2. being relatively maximal corresponds to being a geodesic ray in the sense of [BBJ]; 3. being in E 1 (L) corresponds to having brewise nite-energy and linear function with logarithmic Example 5.2.4.3 (Relative dimension zero, part 2). We now consider what
	growth; rithmic growth; will be a model case for many future considerations: we still work in relative dimension zero over D such a way does not!
	4. nally, Ê1 (L) corresponds to nite-valued harmonic functions with log-arithmic growth. It well-known that any harmonic function on the punctured disc decomposes as a sum of a multiple of log |z| and the real part of an analytic function. This is where our general setting starts diverging from the better-behaved S 1 -invariant. Indeed, by [BBJ, Proposition 4.1], for rays of metrics of nite This is a nice model case for us, because the Deligne pairing construction (in our setting of brations over D Corollary 5.2.4.4. The relative maximality condition for metrics in E 1 (L) can be pushed forward to the base via the Deligne pairing, i.e. we have a well-dened map
	energy, maximality implies linear growth. However, in our case, maximality

* , L M × D * ).

4. therefore, belonging to Ê1 (L) corresponds to being a brewise niteenergy geodesic rays with linear growth emanating from a given point -exactly the space of rays R 1 (L) considered in [DL20]. Example 5.2.4.2 (Relative dimension zero, part 1). Consider the case of relative dimension zero with a trivial line bundle L over X D * . Then, 1. PSH(L) corresponds to the set of subharmonic functions with logarithmic growth on D * ; 2. the class of relatively maximal metrics in PSH(L) corresponds to the class of harmonic functions; 3. E 1 (L) corresponds to nite-valued subharmonic functions with loga-* , but we choose a nontrivial line bundle on D * . The existence of a model for (D * , L) means that there is a relatively ample extension L → D. We can now pick a trivialization τ of L over D, which allows us to identify a metric φ ∈ PSH(L) (extended to L via the logarithmic growth condition!) with the function u = -log |τ | φ on D. By the discussion above, if dd c φ = 0, then u decomposes as u(z) = c • log |z| + H(z), where H is bounded on D. * ) naturally gives line bundles over D * , as we see in action now.

  φ 1,z )+(a 02 +a 21 ) log |z|. taking (negative) Lelong numbers and adding constants, we nda 02 + a 21 -ν 0 (d 1 (φ 0,z , φ 1,z ) + (a 02 + a 21 ) log |z|) ≤ a 02 -ν 0 (d 1 (φ 0,z , φ 2,z ) + a 02 log |z|) + a 21 -ν 0 (d 1 (φ 2,z , φ 1,z ) + a 21 log |z|).

	Upon

Since

z → d 1 (φ 0,z , φ 1,z ) + (a 02 + a 21 ) log |z| is bounded above, the previous equation is by the very denition of d 1 equivalent to

elle préserve les propriétés topologiques de connexité, séparation, compacité;

2. elle réalise une équivalence de catégories entre les catégories de faisceaux cohérents sur X et X

an ; 3. elle s'étend aux variétés sur C, auquel cas elle coïncide avec l'analytication usuelle au sens de Serre.

very ample if L is π-very ample;

The upshot is that this problem gives, intuitively, a purely non-Archimedean criterion for the existence of a family of complex manifolds degenerating to a Kähler-Einstein manifolds! (Of course, the same problem arises in the (possibly twisted) Fano case.) Finally, we briey mention an additional diculty in the Calabi-Yau case. By a counterexample of Cao-Guenancia-Paun, we know that a family φ = (φ z ) z

Now, for all m, by [BJ18a, Lemma 3.1], d 1 (N m (φ 0 ) ∨ N m (φ), N m (φ 1 ) ∨ N m (φ)) ≤ d 1 (N m (φ 0 ), N m (φ 1 )), which at the limit and using the equalities above yields d 1 (N • (P (φ 0 , φ)), N • (P (φ 1 , φ))) ≤ d 1 (N • (φ 0 ), N • (φ 1 )), i.e. by denition d 1 (P (φ 0 , φ), P (φ 1 , φ)) ≤ d 1 (φ 0 , φ 1 ), as promised.

In order to prove that d 1 satises the triangle inequality, and also to make some later results easier to prove, we will approximate the d 1 distance as follows. We approximate two metrics φ 0 and φ 1 in E 1 (L an ) by sequences (φ k 0 ), (φ k 1 ) in C 0 (L an ) ∩ PSH(L an ). We will show that

Proposition 2.4.3.8. Given two metrics φ 0 , φ 1 ∈ E 1 (L an ), and nets (φ k 0 ), (φ k 1 ) in C 0 (L an ) ∩ PSH(L an ) decreasing to φ 0 , φ 1 we have

Proof. By [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF]Remark 5.4.5], i.e. the Darvas formula for d 1 on continuous psh metrics, we have for all k d 1 (φ k 0 , φ k 1 ) = E(φ k 0 , P (φ k 0 , φ k 1 )) + E(φ k 1 , P (φ k 0 , φ k 1 )).

P is continuous along monotone (hence decreasing) nets, so that P (φ k 0 , φ k 1 ) decreases to P (φ 0 , φ 1 ) ∈ E 1 (L an ), and the result follows by continuity of the energy along decreasing nets.

We may now show that E 1 (L an ), endowed with d 1 , is a metric space.

Proof of Theorem 2.4.3.4. Symmetry is immediate. The triangle inequality follows from Proposition 2.4.3.8 and the triangle inequality of d 1 on continuous psh metrics, so that we only have to show that our distance does indeed separate points.

Assume rst that φ 0 ≥ φ 1 , so that the distance is in fact a Monge-Ampère energy. Then, Proposition 4.2.6.2 gives φ 0 = φ 1 .

In the general case, we use Corollary 2.4.3.8 to nd 0 = d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 )).

Both quantities on the right-hand side are positive, which yields φ 0 = P (φ 0 , φ 1 ) = φ 1 , by the previous argument, proving our result.

Okounkov bodies associated to section rings.

Throughout this Subsection, we follow [START_REF] Boucksom | Corps d'Okounkov[END_REF] and [START_REF] Kaveh | Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory[END_REF]. We begin with the following denition. Denition 3.1.2.1. A valuation with one-dimensional leaves with values in an totally ordered group (G, <) on R(X, L) is a valuation ν : R(X, L) → G, such that for all α ∈ G, and all positive integers k, the quotient vector spaces (or leaves) gr k,α (R(X, L)) := {s ∈ H 0 (kL), ν(s) ≥ α}/ α >α {s ∈ H 0 (kL), ν(s) ≥ α } have either dimension zero or one. In fact ([KK12, Proposition 2.4]), each graded piece H 0 (kL) decomposes as a sum of nitely many such leaves.

We will mostly be interested in valuations with value group Z d . Denition 3.1.2.2. Let ν be a valuation with one-dimensional leaves, taking values in (Z d , <) for some total order < on Z d . We dene the sub-semigroup Γ(H 0 (kL)) ⊆ N d of possible values of ν:

Γ(H 0 (kL)) = ν(H 0 (kL)), and nally the semigroup of integral points of (R(X, L), ν), denoted by Γ(R(X, L)) as the graded sub-semigroup of N d+1 dened as follows:

It then follows that (k, α) ∈ Γ(R(X, L)) if and only if dim gr k,α (R(X, L)) = 1.

For brevity, we will write Γ(R(X, L)) = Γ(L).

Note that it still depends on the choice of a valuation. Lemma 3.1.2.3 ([Bou12, L2.11, P3.3]). The semigroup Γ(L) satises the following properties: Theorem 3.2.2.1. Let (X, L) admit continuity of envelopes, with L ample. The asymptotic Fubini-Study operator FS then denes a bijection

The proof follows that of the aforementioned theorem. We start with preparatory lemmas: Lemma 3.2.2.2. Assume that ζ • ≥ ζ • pointwise. Then,

Proof. We rst notice that, since ζ • ≤ ζ • pointwise, the denition of d 1 using successive minima implies Proof. The rst assertion follows from [BE,L7.23] (and its proof). To show asymptotic equivalence, by the previous lemma, it is thus enough to show that FS(N • (FS(ζ • ))) = FS(ζ • ). But, by [BE,T7.26],

which in turn is equal to FS(ζ • ) itself, since it is a limit of an increasing net of Fubini-Study potentials.

We now prove Theorem 3.2.2.1.

Proof. Note that

and by Theorem 3.2.1.1, the right-hand side is in fact equal to

The trick is now to prove the following:

where ∧ denotes the min operator. Since, by Lemma 3.2.2.3,

and the same holds for ζ • , then

and furthermore, by [START_REF] Boucksom | A non-Archimedean approach to K-stability[END_REF](4 [BE,T7.26]. We may now rewrite (3.5) as:

so that the two energies above have the same sign. In particular, the distance

Chapter 4

Geodesics in non-Archimedean pluripotential theory.

Summary of the main results.

This Chapter focuses on the main results of [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF]. Namely, we dene a class of non-Archimedean plurisubharmonic segments, mimicking the a posteriori characterization thereof in the complex case, and we prove the following: Theorem 4.0.0.1. Given φ 0 , φ 1 ∈ E 1 (L), we set d 1 (φ 0 , φ 1 ) = E(φ 0 , P (φ 0 , φ 1 )) + E(φ 1 , P (φ 0 , φ 1 ))

. Then, 1. (E 1 (L), d 1 ) is a metric space; 2. there exists a maximal psh segment t → φ t joining φ 0 and φ 1 ; 3. φ t ∈ E 1 (L) for all t; 4. the segment φ t is a (constant speed) metric geodesic for d 1 , i.e. there exists a real constant c ≥ 0 such that

Denition 4.1.1.3. A plurisubharmonic segment or psh segment is a map [0, 1] → PSH(L) which is a decreasing limit of a net Fubini-Study segments.

Proposition 4.1.1.4. The class of psh segments is the smallest class of segments

which contains all segments of the form

for s a section of some kL and λ, λ ∈ R, is stable under nite maxima, addition of constants, and decreasing limits of nets.

Proof. If we can show that the set of psh segments on L satises all those properties, then it will by denition be the smallest such class. As was the case for the proof of the same result for psh metrics rather than segments, only the property of being stable under decreasing limits is not immediate from the denition. However, using the trick from the proof of [BJ21, Proposition 5.6(vi)], we can reduce to the case of a decreasing net of Fubini-Study paths, and by stability under maximum we may also assume that our net (φ t,α ) α only contains segments of the form

Since our segments are assumed to be decreasing along the net, xing t gives a decreasing net (φ t,α ) α of L-psh metrics, which then converges to a L-psh metric φ t . Therefore, for all t ∈ [0, 1], φ t is not identically -∞. The problem is that we do not know whether the nets of constants converge to nite values.

But taking t = 0, 1 yields in particular that the nets k -1 α (log |s α | + λ α ) and k -1 α (log |s α | + λ α ) decrease to the L-psh metrics φ 0 and φ 1 . Let x be a point on which φ 0 and φ 1 are nonsingular. Then,

is nite, and constant on the set of all such x. Performing this argument for all pairs a < b ∈ [0, 1] shows that φ t corresponds to the segment

which is a psh segment, as desired.

where (s i ) and (t i ) are basepoint-free bases of H 0 (kL). By monotonicity of norm geodesics in the form of Proposition 1.2.5.4, it is enough to show that

but by [BE,Lemma 7.23(i),(iii)] and anti-monotonicity of the N k operator, we have

and similarly for ζ 1 , which proves the result.

Maximal psh segments.

We conclude this section by stating a central Theorem in this article:

Theorem 4.1.3.1. Let φ 0 , φ 1 be any two psh metrics on L. Then,

• either there exists no psh segment between φ 0 and φ 1 ,

• or there exists a unique maximal psh segment t → φ t between φ 0 and φ 1 .

We will prove this result in Section 4.3.2. In what follows, we will state and prove versions of this result in larger and larger classes of metrics, starting from the continuous psh case, then nite-energy metrics, and nally general psh metrics. In each case, we show that the maximal segment remains in the same class as the endpoints, for all t.

4.2 Geodesics between continuous psh metrics.

Throughout this Section, we assume that L is an ample line bundle over a projective K-variety X, K non-Archimedean, and that continuity of envelopes holds for (X, L).

Main Theorem for continuous psh metrics.

We start by studying maximal psh segments in the space of continuous psh metrics. The main Theorem of this section is then the following:

Theorem 4.2.1.1. Let φ 0 , φ 1 be two continuous psh metrics on L. Then, 1. there exists a (unique) maximal psh segment (t, x) → φ t (x) joining φ 0 and φ 1 ; 2. this segment is continuous in both variables; 3. this segment is a geodesic segment for the distance d 1 ; 4. the Monge-Ampère energy is ane along this segment, and it is the unique psh segment joining φ 0 and φ 1 with this property.

A non-Archimedean Kiselman minimum principle.

In this Subsection, we prove an auxiliary result, of independent interest, that will help us prove the rst two points of Theorem 4.2.1.1.

Remark 4.2.2.3. We have stated our minimum principle so as to match the form it will be used in, in the next subsection. A brief look at the proof shows that it can be generalized to the following statement: given a Fubini-Study "polyhedron" (or psh, upon taking decreasing limits) parameterized as Proof. We start with continuity. Since φ 0 , φ 1 are continuous, by continuity of envelopes P (φ 0 , φ 1 -τ ) is continuous for all τ as well. Start by choosing a compact interval S = [a, b] ⊂ (0, 1).

• for large positive τ , and for all t ∈ S, φ 1 -τ ≤ φ 0 (since φ 0 , φ 1 are continuous, thus bounded) and tτ + P (φ 0 , φ 1 -τ ) = tτ + P (φ 1 -τ ) = P (φ 1 ) + (t -1)τ.

Since t -1 < 0, (t -1)τ is very negative while P (φ 1 ) is bounded, so that tτ + φτ does not contribute to the supremum; 4.2.6 A result concerning comparable metrics with zero relative energy.

We have used, in the proof of Theorem 4.2.1.1, the fact that if two comparable metrics have the same Monge-Ampère energy, then they are equal. The most natural setting for this result is that of nite-energy metrics, and indeed we will use it in its full generality in the proof of Theorem 4.3.1.1. In this section, we prove this result. We will need another bifunctional acting on continuous psh metrics, the I energy.

Denition 4.2.6.1. Let φ 0 , φ 1 be two continuous L an -psh metrics. Their relative I-energy is dened as

Their relative J-energy is dened as

Given a reference metric φ ref , we write

By [BJ21, Section 6.3], the functionals I and J also admit an extension to E 1 (L an ), which is continuous along decreasing nets.

Note that we have by [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF](3.15)]

so that the I-energy is always nonnegative. This result relies on a special case of the local Hodge index theorem, as in [BJ21, Proposition 3.5]. That the Jenergy is nonnegative follows from the very expression of the Monge-Ampère energy.

Proposition 4.2.6.2. Let φ 0 , φ 1 ∈ E 1 (L an ). If E(φ 0 , φ 1 ) = 0 and φ 0 ≥ φ 1 , then φ 0 = φ 1 .

Proof. The main argument has been communicated to the author by S.

Boucksom and M. Jonsson, as part of works on nite-energy spaces currently in writing.

Approximate φ 0 and φ 1 by decreasing nets φ k 0 , φ k 1 ∈ C 0 (L an ) ∩ PSH(L an ). Up to taking the maximum of the two sequences, we can assume without loss of generality that for all k, φ k 0 ≥ φ k 1 . We have that

Since φ 0 ≥ φ 1 , all of the terms in the above sum are integrals against positive measures of nonnegative functions, hence they are all positive. In particular,

where the vanishing follows from continuity of E along decreasing nets, and the fact that E(φ 0 , φ 1 ) = 0.

Pick any positive measure µ that can be expressed as MA(φ) for some φ ∈ C 0 (L an ) ∩ PSH(L an ), and write for x ∈ X an

In our case, we then have

141 recalling that we have dened µ = MA(φ). Now, by continuity of the extensions of I and J along decreasing nets,

while we have established before that

We then nd that

and the right-hand side vanishes, while the left-hand side converges as k → ∞

to the nonnegative quantity

The key point now is to solve the non-Archimedean Monge-Ampère equation in order to nd a measure µ = MA(φ) with positive Dirac mass at x. Now, we recall that a psh function is uniquely determined by its restriction to the set of divisorial points in X an , and that for any such point x we may nd a Monge-Ampère measure µ x associated to a projective model of X which has an atom at x, as in [BE,Example 8.11]. As we then have

and µ x ({x}) > 0, we have that φ 0 = φ 1 on all divisorial points of X an , hence on X an .

A simple example.

Consider the symmetric algebra Sym • (V ) of some nite-dimensional K-vector space V . One can easily see that the k-th symmetric power of a geodesic

Using this result together with our quantization results, we have a nice example: on projective spaces, Fubini-Study segments are in fact maximal psh segments.

Corollary 4.2.7.1. Let X = P n for some positive integer n and L = O P n (m) for some positive integer m. Let ζ t be a norm geodesic in H 0 (L). Then, the psh geodesic

This does not hold in general: even if the boundary norms are generated in degree m for some m, the geodesic is not necessarily in degree m for all t,

and thus the result only holds "at innity".

Proof

in H 0 (kL) for all k through the k-th symmetric powers, as discussed above. Therefore, for all t, the geodesic is generated in degree one, and we have

proving our result.

4.3 Geodesics between nite-energy metrics.

Main Theorem for nite-energy metrics.

Our main Theorem in this Section is the following.

Theorem 4.3.1.1. Given φ 0 , φ 1 in the metric space (E 1 (L), d 1 ), we have that:

1. there exists a maximal psh segment t → φ t joining φ 0 and φ 1 ; 2. φ t ∈ E 1 (L) for all t; 3. the segment φ t is a (constant speed) metric geodesic for d 1 , i.e. there exists a real constant c ≥ 0 such that

for all t, s ∈ [0, 1]; 4. the Monge-Ampère energy is ane along φ t , and it is the unique psh segment joining φ 0 and φ 1 with this property.

and similarly for ψ k t . Both nets converge to the limit sup{ϕ t psh segment between φ 0 and φ 1 }, which depends only of the endpoints φ 0 and φ 1 , proving the Theorem.

4.3.3

Proof of Theorem 4.3.1.1.

From the previous section, the unique maximal psh segment joining two nite-energy metrics φ 0 , φ 1 can be recovered as the limit φ t of maximal segments φ k t joining decreasing approximations φ k 0 , φ k

It could a priori be the case that this leaves the class E 1 (L). Theorem 4.3.1.1(3) will ensure that this is not the case.

Proof of Theorem 4.3.1.1 (2)-( 5). As we have just discussed, existence (i.e.

(2) in the Theorem) is ensured by Theorem 4.1.3.1. For all k, and for any reference metric

. By continuity of the energy along decreasing nets, the limit function

is therefore ane, with coecient equal to the (nite) energy E(φ 0 , φ 1 ). This gives (3).

Furthermore, by Proposition 4.2.6.2, that φ t is the only possible psh segment with the property that the Monge-Ampère energy is ane along it is proven using the same arguments as the proof of Theorem 4.2.1.1(4), establishing

(5). To show that they are geodesics for our extended d 1 distance on E 1 (L) again follows from the fact that the segments φ k t are d 1 -geodesic, by Theorem 4.2.1.1: for all t, t ,

and taking the limit in k, using Proposition 2.4.3.8.

Chapter 5

The space of nite-energy metrics over a degeneration.

Summary of the main results.

This Chapter covers the results of [START_REF] Reboulet | The space of nite-energy metrics over a degeneration of complex manifolds[END_REF], which is concerned with generalizing the formalism of plurisubharmonic geodesic rays to the case of arbitrary (non-isotrivial) degenerations over the punctured unit disc, and studying non-Archimedean interpretations, building on the work of Darvas-Lu ([DL20]) and Berman-Boucksom-Jonsson ( [BBJ]).

At least half of this Chapter takes place in the complex world; we have observed silence regarding complex pluripotential theory in this manuscript, rather focusing on the non-Archimedean case. Whenever necessary (and indeed this is where we shall begin), we will recall relevant notions in their right context.

5.1 Relative nite-energy spaces.

5.1.1 Reminders on nite-energy spaces in complex geometry.

We begin with some reminders concerning d 1 -structures on spaces of niteenergy metrics in the classical setting. We thus consider a xed compact Kähler manifold X, with dim X =: d, endowed with an ample line bundle L.

this convergence to hold, one sees that this construction is multilinear, symmetric, satises the change of metric formula. Uniqueness follows from the change of metric formula, which itself shows that the construction glues well over X.

That it would dene a nite-valued metric on U follows from Lemma 5.1.2.3 below, so that all that is left in order to prove the Theorem is that the limit in question is decreasing. We proceed by induction on the number n of indices i ∈ {0, . . . , d

In the case n = 0, all metrics are continuous psh and this is the classical Deligne pairing, so that we have nothing more to prove.

Assume thus that the assertion holds for some d + 1 > n > 0. Assume the metrics φ i , i = 1, . . . , d + 1 -n to belong to C 0 ∩ PSH(L i ), and the n + 1 other metrics φ i to belong strictly to E 1 X/U (L i ), i = 0 or i = d + 2 -n, . . . , d.

(We can do this without loss of generality, by symmetry and up to reordering the indices.) We approximate φ 0 and the (φ i ) d i=d+2-n by sequences k → φ k i of continuous psh metrics. For a xed ∈ N * and by the induction assumption, the sequence

is decreasing and converges to a limit φ 0 , φ 1 , . . . , φ d+1-n , φ d+2-n , . . . , φ d X/U . This limit satises, for any xed metric φ 0 ∈ C 0 ∩ PSH(L 0 ) the formula

Now, this expression yields a decreasing net as increases, and its limit is nite. In particular, it be seen to be the decreasing limit of

which proves our desired statement by induction.

Along the way, we have used the following Lemma regarding niteness of products of absolute nite-energy classes. The proof follows from exactly the arguments of [BJ21, Theorem 5.8], therefore we leave the details to the interested reader.

Lemma 5.1.2.3. Let X be a compact Kähler manifold of dimension d, and let (L i ) be a collection of d + 1 ample line bundles on X. Fix, for all i = 0, . . . , d, a metric φ i ∈ E 1 (L i ), and a continuous metric φ 0 ∈ E 1 (L 0 ). Then, the integral

5.1.3 Relative plurisubharmonic segments.

In the case where Y is a point (the absolute setting), it is well-known ( [START_REF] Darvas | The Mabuchi geometry of nite energy classes[END_REF]) that any two metrics φ 0 , φ 1 in E 1 X/Y (L) = E 1 (X, L) can be joined by a plurisubharmonic geodesic segment in E 1 (X, L), in the following sense.

There exists a S 1 -invariant plurisubharmonic metric Φ on the product L × A (where A is the annulus {e

such that Φ bounds by above all other such segments Ψ with Ψ 0 ≤ φ 0 and Ψ 1 ≤ φ 1 . We now look at what happens when Y is no longer a point. Given φ 0 , φ 1 ∈ E 1 X/Y (L), and a point y ∈ Y on the base, there exists by the previous discussion a plurisubharmonic geodesic segment t → φ t,y joining φ 0,y and φ 1,y in E 1 (X y , L y ). Varying y, this gives a collection of plurisubharmonic geodesic segments t → φ t . It is not obvious that, for given t, φ t has plurisubharmonic variation with respect to Y . We thus claim the following: Theorem 5.1.3.1. Given any φ 0 , φ 1 ∈ E 1 X/Y (L), the collection [0, 1] t → φ t of brewise psh geodesic segments belongs to E 1 X/Y (L). Furthermore, identifying the collection t → φ t with a S 1 -invariant metric Φ on L × A, Φ is plurisubharmonic, and is the unique psh metric on L × D * such that for all y ∈ Y ,

where π y : X × A → {y} × A is the projection to the point y. Such a segment will be called a psh geodesic segment in E 1 X/Y (L).

The last statement can be interpreted as saying that the Monge-Ampère energy along the psh geodesic segment is brewise ane.

then Pφ is continuous up to the boundary.

2. Construction of a subsolution under the hyperconvexity assumption.

The second point of the Theorem will follow from a more general principle:

consider the class C(L| ∂M ) consisting of continuous, brewise psh metrics on L| ∂M admitting a continuous psh extension (hence a relatively maximal continuous extension, by the rst point) to all of L| M . Then, this class is stable under uniform limits, which follows from seeing that the mapping C(L| ∂M ) φ → Pφ from the rst part of the proof is continuous under uniform convergence.

To prove the second point, we therefore have to show that there exists a sequence φ k ∈ C(L| ∂M ) converging uniformly to our boundary data φ. We proceed bu Bergman kernel approximation. Since L is π-ample, the sheaves π * (kL) are locally free for all k large enough, and correspond to the sections of a vector bundle E k whose bres are the H 0 (kL z =, z ∈ M . The collection of L 2 -norms N k (φ) associated to kφ then dene a continuous collection of Hermitian metrics h k on E k | ∂Ω . We pick a sequence of smooth families of Hermitian metrics (h k,j ) j on π * (kL) so that h k,j → h k uniformly on π * (kL)| ∂Ω . The associated collection of metrics

vary smoothly on L. Since they are brewise smooth and strictly psh (both of which are necessary conditions for the following argument), we may compensate for the lack of plurisubharmonicity in the direction of z, by pulling back a high enough multiple m k,j π * ρ of the dening function ρ of Ω, which as we recall vanishes on the boundary of Ω. We therefore have a continuous psh extension

. Now, by Bergman kernel approximation, the φ k themselves converge uniformly and increasingly to φ, which implies φ ∈ C(L| ∂M ), concluding the proof.

Remark 5.1.4.6. By adapting classical arguments of pluripotential theory, one shows that a continuous psh metric on L is relatively maximal i (dd c φ) d+1 = 0 on X, i it coincides over each relative open subset with its Perron-Bremmermann envelope as in the above Theorem.

We now characterize relatively maximal metrics of relative nite energy.

Proposition 5.1.4.7. Let φ be a metric in E 1 X/Y (L). Assume that Y is covered by relatively compact hyperconvex smooth open subsets. Then, φ is relatively maximal if and only if φ d+1 X/Y has zero curvature. Proof. Assume φ to be relatively maximal. Since the Deligne pairing construction is stable upon restriction to an open set, we will work over the preimage U of some smooth hyperconvex relatively compact open set in Y . If φ is continuous, we have just mentioned that (dd c φ) d+1 = 0 there, so that by (5.2), it follows that dd c E(φ) ≡ 0. The non-continuous case follows from regularization on U : pick a sequence of continuous metrics φ k decreasing to φ on U ; by Theorem 5.1.4.4, there exists a continuous, relatively maximal psh metric Φ k coinciding with φ k on L| π -1 (∂U ) . By maximality, the sequence Φ k necessarily converges to φ (since φ is assumed to be relatively maximal), and continuity of Deligne pairings along decreasing nets ensures φ d+1 X/Y to have zero curvature.

Conversely, assume φ d+1 X/Y to have zero curvature. In the continuous case, using (5.2) again, it follows that (dd c φ) d+1 = 0, as it is a nonnegative measure. In the general case, we again proceed base-locally, and approximate φ on the preimage of a relatively compact open subset U via a decreasing sequence of continuous psh metrics k → φ k . Let Φ k be for each k the unique continuous and relatively maximal metric on U with prescribed boundary condition φ k | π -1 (∂U ) , given by Theorem 5.1.4.4. Let Φ denote the limit of the decreasing sequence k → Φ k , which is relatively maximal. By continuity of the Deligne pairing along decreasing nets, this sequence also denes a decreasing sequence of zero curvature metrics Φ d+1 k X/U which has to converge to the metric Φ d+1 X/U , which is a zero curvature metric φ on U , coinciding on ∂U with φ d+1 X/U . Since φ d+1 X/U also has zero curvature, we have to have

on all of U . Fix z in U , and note that this implies

while by relative maximality of Φ, φ z ≤ Φ z , which implies Φ z = φ z , thus concluding our proof.

5.2 Finite-energy metrics over degenerations.

5.2.1 Analytic models and degenerations.

We now turn to our main setting. We will consider the base Y to be the punctured unit disc, and we will assume that our family degenerates (meromorphically) as one approaches zero.

Denition 5.2.1.1. Consider a holomorphic submersion π : X → D * with compact Kähler bres, and a relatively ample line bundle L on X. An analytic model (or simply a model) of X is a normal complex analytic space X , together with a at, proper holomorphic morphism π : X → D, realizing an isomorphism X π -1 (D * ). An analytic model of (X, L) is the data of an analytic model X on X, and an ample line bundle L over X such that L restricted to π -1 (D * ) is isomorphic to L. We dene a degeneration (or a degeneration with meromorphic singularities) to be a morphism π : X → D * as above, such that there exists an analytic model of (X, L).

Example 5.2.1.2. This construction specializes to the following well-known cases:

• if all the bres of X are isomorphic to M , a model X can simply be viewed as a compactication of an isotrivial degeneration of M ;

• if the above condition holds, and furthermore the isomorphism is generated by a C * -action, this is simply a (real) one-parameter degeneration of (M, L| M ), i.e. a test conguration for (M, L| M ).

The central bre of a model of X is the space X 0 = π -1 ({0}). If the degeneration X → D * is isotrivial, we say that M , the bre over 1, is the generic bre of X.

Generalized slopes and Lelong numbers.

As we will be working with (generalized) subharmonic functions on the base D * , we will often have to work with some notions of Lelong numbers. We review some (old and new) facts in this Section.

Denition 5.2.2.1. We say that a subharmonic function f on D * has logarithmic growth (near zero) if there exists a real number a such that f (z) + a log |z| is bounded above near zero.

Furthermore, a metric φ ∈ E 1 (L) belongs to Ê1 (L) if and only if, for any model (X , L) of (X, L) and any trivialization of the Deligne pairing L d+1 X /D , denoting u = -log |τ | φ d+1 X /D , one has

where c is a real constant and H is a harmonic function on D depending only on τ and the boundary data.

Proof. The map above is naturally given by φ → φ d+1 X/D * , in which case both statements are corollaries of Proposition 5.1.4.7 and the two examples above.

Metrization.

As an important, and somewhat surprising consequence of our previous results, we may dene a metric structure on the space Ê1 (L). This generalizes e.g. [START_REF] Darvas | Geodesic stability, the space of rays and uniform convexity in Mabuchi geometry[END_REF], in which the authors endow the space of maximal psh rays with the distance d1 (φ 0 , φ 1 ) = lim t→∞ d 1 (φ 0,t , φ 1,t ) t .

In the next Sections, we will show that this structure furthermore satises some good properties, namely completeness and geodesicity.

Theorem 5.2.5.1. The space Ê1 (L) can be endowed with a metric space structure, dened by the generalized slope d1 (φ 0 , φ 1 ).

Naturally, this suggests that the d 1 -distance is subharmonic with logarithmic growth along metrics in Ê1 (L), a fact that we prove now.

Proposition 5.2.5.2. Let φ 0 , φ 1 ∈ Ê1 (L). Then, the map

is subharmonic with logarithmic growth on D * .

Proof. By the formula for d 1 ,

-2 P (φ 0,z , φ 1,z ) d+1 .

By Proposition 5.1.4.7, the rst two metrics on the right-hand side have zero curvature, therefore we are left to show that the metric P (φ 0 , φ 1 ) d+1 X/D * is superharmonic. We pick any zero curvature metric φ ref on D * , and note that P (φ 0 , φ 1 ) d+1 X/D * is superharmonic if and only if P (φ 0 , φ 1 ) d+1 X/D * -φ ref is a superharmonic function. Fix a ∈ D * and let r > 0 be such that D(a, r) = {|z -a| ≤ r} ⊂ D * . Let ψ be the relatively maximal psh metric on D(a, r) and with boundary data φ(z) = P (φ 0,z , φ 1,z ), ∀z ∈ S(a, r).

(5.4) Such a metric is given by Theorem 5.1.4.4. We now deduce the two following facts:

(i) by maximality of ψ, if follows from Proposition 5.1.4.7 that z → ψ d+1 X/D * has zero curvature;

(ii) since on the boundary S(a, r) we have ψ(z) ≤ φ 0,z , φ 1,z , and φ 0 , φ 1 are relatively maximal, we have ψ z ≤ φ 0,z , φ 1,z for all z ∈ D(a, r), thus ψ z ≤ P (φ 0,z , φ 1,z ) and nally

by monotonicity of the Monge-Ampère energy.

Using (5.4), (i), and (ii) in order, we nd: S(a,r)

As the inequality is true for all a, our metric P (φ 0 , φ 1 ) d+1 X/D * is then superharmonic.

We now show that there exists a real number a ∈ R such that z → d 1 (φ 0,z , φ 1,z ) + a log |z| is bounded above. By Lemma 5.2.3.2(iv), for any model (X , L) of (X, L), xing a reference metric φ ref ∈ Ê1 (L) which is locally bounded on L, one has (up to adding large enough constants)

and the term on the right-hand side is a harmonic function with logarithmic singularities at the origin, so that substracting constants the result also holds for z → d 1 (φ 0,z , φ ref,z ). Proceeding similarly for φ 1 , our result then follows from the triangle inequality.

Finally, we note an immediate consequence of Lemma 5.2.2.5 together with the previous Proposition 5.2.5.2.

Lemma 5.2.5.3. Let φ 0 , φ 1 ∈ Ê1 (L). Then, for all z on the base, we have

Remark 5.2.5.4. Had we not xed boundary data, we would have an additional error term in the above expression, corresponding exactly to the

We are now equipped to endow the space Ê1 (L) with a metric structure.

Proof of Theorem 5.2.5.1. That d1 (φ, φ) = 0 and d1 (φ 0 , φ 1 ) = d1 (φ 1 , φ 0 ) are immediate statements, and nonnegativity will follow from the triangle inequality and the former statement. Therefore, we must show that for any other φ 2 ∈ Ê1 (L), we have d1 (φ 0 , φ 1 ) ≤ d1 (φ 0 , φ 2 ) + d1 (φ 2 , φ 1 ).

Proposition 5.2.6.3. Let φ k be a sequence of metrics in Ê1 (L) converging to some metric φ on L for the topology of base-locally uniform brewise d 1 convergence. Then, φ belongs to Ê1 (L).

Proof. Pick a sequence k → φ k ∈ Ê1 (L) and a xed metric φ in E 1 (L).

Assume that, for a relatively compact open U ⊂ D * we have

Since convergence in Monge-Ampère energy is subordinate to d 1 -convergence we have that

→ φ d+1 z again uniformly in z; by maximality, the metrics φ d+1 k X/D * are zero curvature, and an uniform limit of such is again zero curvature. As having zero curvature is a local property and the π -1 (U ) cover X, we then have that φ d+1 X/D * has zero curvature on all of X. By virtue of being in E 1 (L), this implies φ to be relatively maximal by 5.1.4.7, as long as we can show that the limit is psh. On π -1 (U ), there exists c >

against a xed smooth family of volume forms z → µ z , so that uniform brewise d 1 -convergence implies L 1 convergence of φ k to φ on π -1 (U ), which establishes plurisubharmonicity of the limit there, hence on X.

We may now prove completeness.

Proof of Theorem 5.2.6.1. Consider a Cauchy sequence m → φ m ∈ Ê1 (L).

For all ε and all large enough m, n, is therefore a metric in Ê1 (L) by Proposition 5.2.6.3.

Geodesics.

We now show that, much as in the absolute E 1 setting, one can nd geodesics in Ê1 (L).

Theorem 5.2.7.1. Given any φ 0 , φ 1 ∈ Ê1 (L), the psh geodesic segment t → φ t joining them, given by Theorem 5.1.3.1, is d1 -geodesic in the metric sense, i.e.

Furthermore, given any model (X , L) of (X, L) and a trivialization τ of L d+1 over D, setting

the segment of generalized slopes t → ût is ane on [0, 1]; and t → φ t is uniquely characterized by this property among psh segments. Proof of Theorem 5.2.7.1. Let φ 0 , φ 1 ∈ Ê1 (L). We consider as in Theorem 5.1.3.1 the family of brewise maximal geodesics t → φ t,z .

To show that it belongs to Ê1 (L), we must make sure that it has logarithmic growth and is relatively maximal. The former is due to Lemma 5.2.3.2(ii), since for xed x ∈ X, φ t (x) ≤ (1 -t)φ 0 (x) + tφ 1 (x) by convexity of maximal segments, so that if there exist a i , i = 0, 1 such that φ i +a i log |z| are bounded above near the central bre of some model, then so is φ t + (1 -t)a 0 + a 1 . Regarding maximality, φ d+1 t X/D * is a convex combination of zero curvature metrics with logarithmic growth, hence φ t is also relatively maximal by Proposition 5.1.4.7.

That t → φ t is d1 -geodesic is a consequence of the fact that, for all z, t → φ t,z is d 1,z -geodesic. Finally, the statement regarding the Monge-Ampère energy follows upon taking generalized slopes in the statement of Theorem 5.1.3.1.

Extension of the distance to E 1 (L)

In this Section, we construct a "maximal envelope" map, which will allow us to extend the d 1 -distance as a pseudodistance to all of E 1 (L).

Proposition 5.2.8.1. For all φ ∈ E 1 (L), there exists a unique smallest relatively maximal metric P (φ) ∈ Ê1 (L) with φ ≤ P (φ) and

as z → 0. This denes a natural projection

Before proving this result, we note this immediate Corollary:

Corollary 5.2.8.2. The mapping d1 (φ 0 , φ 1 ) = d1 ( P (φ 0 ), P (φ 1 ))

denes a pseudodistance on E 1 (L).

Proof of Proposition 5.2.8.1. Let φ ∈ E 1 (L) ∩ C 0 (L), and, for all r ∈ (0, 1), let U r denote the annulus {r < |z| < 1} ⊂ D * , and V r = π -1 (U r ) ⊂ X. Let φ r be the relatively maximal metric on V r , coinciding with φ on ∂V r , given by Theorem 5.1.4.4. Fixing z on the base, the sequence r → φ r,z is an increasing sequence of psh metrics in E 1 (L z ). We claim that the limit family

is the desired envelope P (φ). Denote this limit φ for the moment. Fix some r. By construction, φ restricted to V r coincides everywhere with its Perron-Bremmermann envelope; furthermore, it is locally bounded (since it is approximable from below). By the discussion in Section 5.1.4, since this holds for all r, φ is relatively maximal. Furthermore, by construction again, it satises φ ≤ φ and is the smallest such relatively maximal metric. We are therefore only left to prove that d 1 ( φz , φ z ) = O(log |z|) as z → 0. As in Corollary 5.2.4.4, we pick a model (X , L) of (X, L), and we extend φ d+1 to the trivializable line bundle L d+1 . Picking a trivialization τ allows us to identify the energies φd+1 and the φ d+1 r with functions u and u r on D

Example 5.3.1.1. In the case of an isotrivial degeneration X M × D * for some complex projective manifold M , X can be identied with the base change of M to the eld K. In particular, there exists a "trivial" algebraic model, dened by taking the base change of M to R, which corresponds to the product analytic family over D.

K is a (non-Archimedean) valued eld, with valuation

This also denes a valuation on the Noetherian ring R. From the general work of Berkovich ( [BerkBook]), one can associate to a scheme X over a valued ring R, in a functorial way, its analytication X an with respect to the given valuation on the base. The underlying points of this analytication roughly correspond to valuations on the function eld K(X) extending the base valuation on K, and the topology is that of pointwise convergence.

In our setting, the Berkovich analytication X an K of X K contains an important dense subset: the set of divisorial points X div . It is described as follows. Let X be an analytic model of X. By Noetherianity and normality, the bre of X over 0 is then a Cartier divisor which decomposes as the Weil divisor

with each E i irreducible. Each component of such a decomposition denes a valuation ν E i on K(X) as follows: for all f ∈ K(X),

All divisorial points of X an K are then obtained in this manner.

Relating non-Archimedean psh functions and models.

Let X be a degeneration with a line bundle L on X. Let (X , L) be a model of (X, L). Recall that to L one can associate a model metric φ L on L an K , (as explained, for example, in detail in [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF]). Such a metric is uniquely characterized as follows: given an open set U ⊂ X and a nonvanishing section of the restriction of L to U, then we require that |s| φ L = 1 on (U K ∩ X K ) an .

We thus obtain the following: Corollary 5.3.4.2. For any metric φ ∈ PSH(L), and any model (X , L) of (X, L) such that φ extends as a psh metric on L, there exists an integer m 0 such that the function

is L-psh for all positive integers m. Proof. In the case where φ also has brewise nite energy, this follows from the previous Lemma. In the general case, one can approximate φ on L by a decreasing sequence of (e.g.) locally bounded metrics φ k . Since the integer m 0 depends only on L, the sequence

is then a sequence of L-psh functions. Since the sequence φ k is decreasing, we have for all k that J (φ k+1 ) ⊆ J (φ k ), i.e. the sequence φk is also decreasing, which implies its limit (m + m 0 ) -1 log |mJ (φ)| to be L-psh, as desired.

Proof of Theorem 5.3.3.1.

We may now prove Theorem 5.3.3.1.

Proof. We x a metric φ ∈ PSH(L). We need to show that the function dened on X div by φ NA : ν E → ord E (φ),

where ν E corresponds to a divisorial valuation and ord E is dened as a generic Lelong number as in (5.5), admits a psh extension on X an K . Since a non-Archimedean psh function is uniquely dened on the set of divisorial points, it is then though to show that φ NA can be approximated by a decreasing sequence of psh model functions on X an K . Note that, by construction, the map φ → φ NA is lsc and order preserving. By Corollary 5.3.4.2, the metric

where u m is the model function log |J (mφ)|, is L an R -psh. Pick a divisorial point ν E ∈ X div associated to a component in the central bre of an analytic 1. the radial limit lim t E(φt) t ; 2. the non-Archimedean energy of the non-Archimedean metric φ NA associated to φ.

In [BBJ], it is established that if φ extends to a locally bounded metric on a test conguration, then those two quantities coincide. This is not the case in general, however. In this Section, we generalize those results to our relatively maximal psh metrics on degenerations. It will be clearer to express this using the relative dimension zero case of the construction from the previous Section.

Remark 5.4.1.1 (Relative dimension zero and the non-Archimedean limit). Theorem 5.4.1.2. For all φ ∈ E 1 (L) admitting a locally bounded extension to some model (X , L), we have

As mentioned in

as non-Archimedean metrics on the Deligne pairing L an over Spec K. Proof. Note that the metric φ d+1 X/D * is subharmonic by (5.2), so that the left-hand side is well-dened (this is the relative dimension zero case of Example 5.2.4.3).

We pick a model (X , L) such that φ extends to a locally bounded metric on L. By Proposition 5.3.6.1, we necessarily have φ NA = φ L , the model metric on L an K associated to L, so that we are left to show that, given a trivialization τ of L d+1 Corollary 5.4.2.4. The space E 1 (L) is mapped by (•) NA to E 1 (L an K ); furthermore, for any φ ∈ E 1 (L), we have

In other words, we do not have non-Archimedean extension of the Monge-Ampère energy in E 1 (L), but simply an inequality. Proof. We start by picking a metric φ ∈ E 1 (L), and we dene Φ to be the hybrid maximal metric with Φ NA = φ NA obtained from Theorem 5.4.2.3. Then, since Φ is relatively maximal, φ ≤ Φ, so that by monotonicity of φ → φ NA , ( φ d+1

X/D * ) NA = (Φ NA ) d+1 by hybrid maximality, proving our inequality. To prove the rst statement, it is enough to notice that the logarithmic growth condition built into E 1 (L) forces ( φ d+1 X/D * ) NA to be nite.

The isometric embedding.

We denote by Ê1 hyb (L)

the subspace of hybrid maximal metrics in Ê1 (L). Our main Theorem is the following:

Theorem 5.4.3.1. The inverse of the mapping (•) → (•) NA given by Theorem 5.4.2.3 is an isometric embedding of (E 1 (L an ), d NA 1 ) into ( Ê1 (L), d1 ) with image Ê1 hyb (L). Furthermore, a psh segment in E 1 (L an ) is a psh geodesic if and only if its image is a psh geodesic.

Remark 5.4.3.2. The rst statement of the Theorem can be thought of as saying that hybrid maximal metrics have the d 1 -extension property. The whole of Theorem 5.4.3.1 essentially means that we realize the (non-Archimedean) space E 1 (L an ) as a purely complex geometric object! Proof. We thus rst show that our mapping preserves psh geodesic segments. Pick a psh geodesic segment φ NA t between two metrics φ NA 0 and φ NA 1 in E 1 (L an K ), and consider for all t ∈ [0, 1] the hybrid maximal metric φ t with (φ t ) NA = φ NA t . By Theorem 5.2.7.1, it is enough to show that t → ( φ d+1 t X/D * ) NA is ane; by the Monge-Ampère energy extension property, this is equivalent to asking that t → φ NA t is ane, which holds by [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF]. The reverse implication is proved in the same way.

We now prove the isometry statement. Pick φ 0 , φ 1 in Ê1 hyb (L). We assume both metrics to be continuous, and the general result will proceed as usual from regularization. Using Theorem 5.4.2.3 together with the expressions of the distances and additivity of Lelong numbers,

we only have to show that

Recall that we have seen z → P (φ 0,z , φ 1,z ) d+1 to be superharmonic, so that the left-hand side is well-dened.

Set some r ∈ (0, 1). We consider the relatively maximal metric ψ r on the preimage U r of the annulus {r ≤ z ≤ 1} with boundary data given by z → P (φ 0,z , φ 1,z ) for z ∈ ∂U r , which exists by Theorem 5.1.4.4. Having xed z ∈ X, the sequence r → ψ r,z , r ≤ |π(z)|, is decreasing as r decreases, and therefore the limit lim r→0 ψ r =: ψ is still a relatively maximal metric. As we have, for all |π(z)| = r,

We must now prove that ( ψ d+1 X/D * ) NA = P (φ NA 0 , φ NA 1 ) d+1 .

187

We rst claim that ψ realizes the supremum

Since ψ is itself such a metric, it is enough to show that for all candidates ϕ, we have ϕ ≤ ψ. But for all z ∈ X, since ϕ z ≤ φ 0,z , φ 1,z , we have

and nally

We now conclude: by this extremal characterization of ψ, we have that ϕ NA ≤ ψ NA for all ϕ ≤ φ 0 , φ 1 . In particular, since the construction is orderpreserving, the hybrid maximal metric Ψ with Ψ NA = P (φ NA 0 , φ NA 1 ) satises Ψ ≤ ψ, so that P (φ NA 0 , φ NA 1 ) ≤ ψ NA , while on the other hand, ψ ≤ φ 0 , φ 1 , hence ψ NA ≤ φ NA 0 , φ NA 1 and nally ψ NA ≤ P (φ NA 0 , φ NA 1 ).

Remark 5.4.3.3. The proof of the above result in the case of geodesic rays, which does not appear explicitly in the literature (but is based on some ideas from [BDL]), was nicely explained to the author by Tamas Darvas.

Remark 5.4.3.4. In the above proof, we implicitly dened an envelope operator sending two metrics φ 0 , φ 1 in Ê1 hyb (L) to the largest metric P (φ 0 , φ 1 ) in Ê1 hyb (L) bounded above by φ 0 and φ 1 . In [Xia, Example 3.3], this construction appears already in the case of geodesic rays, and Xia uses this envelope to dene alternative distance

which (a specialization of) our proof shows to coincide with the usual distance d1 . In fact, Xia denes this envelope more generally in [Xia, Example 3.2],

in the radial equivalent of the space Ê1 (L). Proof. We approximate each of the φ NA i by a decreasing sequence of model metrics φ NA i,k , and denote by φ i,k their associated hybrid maximal metrics. By our previous results, φ i,k decreases to φ i by hybrid maximality. Since Deligne pairings are decreasing along (mixed) decreasing limits, using the estimates [BBJ, Lemma A.1, Lemma A.2], we nd for all z in X 0 ≤ φ 0,k,z , . . . , φ d,k,z -φ 0,z , . . . , φ d,z ≤ C(z)

where the slope of C(z), Ĉ, is a nite real constant. Indeed, C(z) is a maximum of a collection of functionals expressed as Deligne pairings, which are all subharmonic along relatively maximal metrics. We take generalized slopes in the above inequality to nd

where we have used the d 1 -extension property of hybrid maximal metrics.

Using Remark 5.4.1.3, we then have that

which has many important norm-like properties and is commonly used to study properties of nite-energy spaces ( [BBEGZ], [START_REF] Boucksom | Uniform K-stability and asymptotics of energy functionals in Kähler geometry[END_REF], see also [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF] for the non-Archimedean trivially-valued case);

2. the J-functional, dened as

which can be seen as a corrected relative Monge-Ampère energy which is translation invariant;

3. the twisted energy functionals, dened as

, where L is another line bundle on X. A special case of it appears in the expression of the Mabuchi K-energy, and the study of its slopes in the trivially-valued case is essential to establish the general (cscK) case of the Yau-Tian-Donaldson conjecture, as in [Li].

5.4.5 Test congurations and the trivially valued case.

All of our previous results encapsulate the trivially valued case, as we explain now. Let π : X → D * be now a polarized test conguration, i.e. a degeneration with relatively ample line bundle L such that π and L are equivariant under some C * -action (forcing all bre pairs (X z , L z ) to be isomorphic). One may then choose a reference continuous psh metric φ ref on the bre at 1 and require our psh metrics φ to satisfy φ z = i z * φ ref for z ∈ S 1 , and with i z : X z → X 1 the isomorphism as mentioned above. The authors in [BBJ] (e.g.) study

the space E 1 0 (X an 1 ) of nite-energy metrics over the analytication of X 1 with respect to the trivial absolute value on C. We denote by R 1 (L 1 ) the space of hybrid maximal nite-energy rays in PSH(L 1 ) emanating from φ ref (where, as mentioned before, a hybrid maximal ray corresponds in the terminology of [BBJ] to a maximal psh geodesic ray). We then claim the following: Proposition 5.4.5.1. There is a sequence of distance-preserving maps

where the rst and last maps are bijective (i.e. isometries), while the middle map is injective.

Proof. The case of the last map has been treated by Theorem 5.4.3.1. The rest of the proof is merely a matter of correctly dening our maps.

For the rst map, the bijection is given by [START_REF] Berman | A variational approach to the Yau-Tian-Donaldson conjecture[END_REF]Theorem 6.6]. The metrization of the space E 1 0 (L an 1 ) is described in a [START_REF] Boucksom | Global pluripotential theory over a trivially valued eld[END_REF], but proceeds much as the metrization of E 1 (L an ) in [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF], while we recall that we metrize the space of maximal rays by d1,0 (φ, φ ) = lim t t -1 d 1 (φ t , φ t ) and take equivalence classes to yield the space R 1 (L 1 ). (We direct the reader to e.g. [BDL]. Note that in the cited article, the authors consider the space of all (non-necessarily hybrid) maximal psh rays.) Proving the distancepreservingness of the isomorphism is then essentially a simpler version of Theorem 5.4.3.1, which we leave to the interested reader.

We claim that the middle map, which we will denote ι 0 , can be represented as follows: let φ : t → φ t be a hybrid maximal psh geodesic ray in X 1 . Let i z be as before the isomorphism i z : X z → X 1 , and dene ι 0 (φ) to be the metric z → i z * (φ -log |z| ).

The distance-preservingness is immediate(!), so that we are left to check that ι 0 (φ) is a hybrid maximal metric. By [BBJ, Corollary 6.7],

t → E(φ t )

is ane, which implies by invariance of the energy under polarized isomorphisms that z → ι 0 (φ)(z)

is harmonic on D * , proving maximality by Proposition 5.1.4.7, and hybrid maximality is given by construction.

Remark 5.4.5.2. One also notices (by mimicking our proofs in the discretely valued case) there to be under the above maps a correspondance {non-Archimedean maximal psh segments in E 1 0 (L an 1 )} {rays of complex geodesics between two rays in R 1 (L 1 )} → {discs of complex geodesics between two metrics in Ê1 hyb (L)} {non-Archimedean maximal psh segments in E 1 (L an )}.

We may state the most interesting part of this result as follows.

Proposition 5.4.5.3. Let t → φ 0,t , t → φ 1,t be maximal psh geodesic rays in the sense of [BBJ]. Let, for all t, [0, 1] s → φ s,t be the maximal psh segment joining φ 0,t and φ 1,t . Then, for all s ∈ [0, 1], t → φ s,t is a maximal psh geodesic ray in the sense of [BBJ]. Furthermore, let for all s ∈ [0, 1], φ NA s be the non-Archimedean metric associated to the psh geodesic ray s → φ s,t . Then, s → φ NA s is the maximal non-Archimedean psh geodesic joining φ NA 0 and φ NA 1 in the sense of [START_REF] Reboulet | Plurisubharmonic geodesics in spaces of non-Archimedean metrics of nite energy[END_REF].

Convexity of non-Archimedean functionals.

Via the isometry ι given by Theorem 5.4.3.1, it is now clear what we mean by "a functional on the space of hybrid maximal metrics", since Ê1 hyb (L) inherits a K-vector space structure by setting, for all φ, ψ ∈ Ê1 (L), and λ ∈ K, φ + λ • ψ = ι -1 (ι(φ) + λ • ι(ψ)).

(In particular, one can see multiplication by a scalar in K as a brewise scaling of the metrics, varying meromorphically.) Denition 5.4.6.1. Let F be a functional on Ê1 hyb (L) and F NA be a functional on E 1 (L an ). We say that F NA is a non-Archimedean extension of F if the diagram

Example 5.4.6.2. By construction, E NA is a non-Archimedean extension of the "discal" energy Ê. Furthermore, all "generalized energy" functionals of Proposition 5.4.4.1 and Example 5.4.4.2 admit non-Archimedean extensions.

By Theorem 5.4.3.1, we get "for free" a way to study convexity of non-Archimedean functionals.

Heuristic 5.4.6.3. Let F be a functional that is convex along complex psh geodesics, F its "discal" version, and F NA a non-Archimedean extension of F . Then F NA is convex along maximal non-Archimedean psh geodesics in E 1 (L an ).

Proof. Given a non-Archimedean maximal psh segment φ NA t joining φ NA 0 and φ NA 1 ∈ E 1 (L an ), we can write using Theorem 5.4.3.1 φ t := ι -1 (φ NA t ) using the maximal psh segments t → φ t,z joining the φ 0,z and φ 1,z . We then simply write for all z F (φ t,z ) ≤ (1 + t)F (φ 0,z ) + tF (φ 1,z ), and take the limit to nd F (φ t ) ≤ (1 + t) F (φ 0 ) + t F (φ 1 ),

which by the denition of a non-Archimedean extension together with ι(ι -1 (φ NA )) = φ NA ) gives F NA (φ NA t ) ≤ (1 + t)F NA (φ NA 0 ) + tF NA (φ NA 1 )

as desired.

Remark 5.4.6.4. Using Proposition 5.4.5.3, the same results also hold mutatis mutandis in the trivially-valued case.