Charles Sébastien

Antoine Chambert- Loir

Antoine Ducros

Walter Gubler

Andreas Höring

Klaus Künnemann

Jérôme

Jian Xiao

Rufei Ren

Xifeng Que

Xujia Zhu

Ao Wang

Chongmo Liu

Xiaoyu Zhang

Xiaojun Wu

Chenlin Gu

Xiaozong Wang

Juanyong Wang

Xiaoqi Xu

Chunhui Liu

Songyan Xie

Zhizhong Huang

Ya Deng

Siarhei Finski

Willie Liu

Huajie Li

Hua Wang

Cheng Shu

Alex Gorentin

Léonardo

Mingkun Gabriel

Yi Liu

Hao Pan

Ratko, Grégoire Mingchen Wu

Xia

ai passé une période inoubliable à l'X et à Diderot avec

Keywords: Espaces de Berkovich, Analyse non-archimédienne, Fibré en droites, Métrique semi-positive, Convexité holomorphe Berkovich spaces, non-Archimedean analysis, line bundles, semi-positive metrics, holomorphic convexity

Ce mémoire de thèse est consacré à l'étude de fibré en droites semipositif en géométrie analytique non-Archimédienne, par un point de vue d'analyse fonctionnelle sur un corps ultramétrique en exploitant la géométrie de la convexité holomorphe.

Le premier chapitre recueille quelques préliminaires pour l'algèbre de Banach sur un corps ultramétique et la géométrie de son spectre au sens de Berkovich, le cadre dans lequel l'étude est effectuée.

Le deuxième chapitre présente la construction de base, qui encode la géométrie intervenante dans certaines algèbres de Banach. On associe une algèbre normée de section à un fibré en droites métrisé. On décrit son spectre, en le reliant avec le fibré en disques unités duals de ce fibré en droites muni de la métrique enveloppante. On encode alors la positivité métrique par la convexité holomorphe.

Le troisième chapitre consiste en deux approaches indépendantes pour le problème d'extension métrique de sections restreintes sur une sous-variété fermée. On obtient une borne supérieure pour la distorsion métrique asymptotique, qui est uniforme par rapport aux choix de sections restreintes. On utilise une propriété particulière aux norms affinoïds pour obtenir cette inégalité.

Le quatrième chapitre traite le problème de la régularité de métrique enveloppante. Avec un nouveau regard venant d'analyse holomorphe à plusieurs variables, on vise à montrer que, quand le fibré en droites est ample, la métrique enveloppante est continue si la métrique de départ l'est. On suggère une méthode tentative reposant sur un analogue non archimédien spéculatif d'un résultat sur la convexité holomorphe due à Cartan et Thullen.

CHAPTER 0 INTRODUCTION (EN FRANÇAIS)

0.1. Motivations 0.1.1. Géométrie analytique non archimédienne. -La géométrie analytique non archimédienne est un analogue de la géométrie analytique complexe. Cette dernière fournit un cadre naturel dans lequel on étudie les fonctions analytiques complexes sur une variété complexe. Dans son analogue non archimédien, le corps des nombres complexes (muni de la valeur absolue usuelle) est remplacé par par un corps valué complet ultramétrique, que l'on appelle un corps non archimédien. Sa naissance répond au besoin d'avoir un bon espace topologique sur lequel on peut définir et opérer les fonctions analytiques à valeur dans un corps non archimédien.

Dans la littérature, plusieurs théories ont été proposées pour formaliser les espaces analytiques non archimédiens, notamment par Tate [START_REF] Tate | Rigid analytic spaces[END_REF], Berkovich [START_REF] Vladimir | Spectral theory and analytic geometry over non-Archimedean fields[END_REF] et Huber [START_REF] Huber | Bewertungsspektrum und rigide Geometrie[END_REF]. Toutes ces théories consistent à recoller des «spectre de Gelfand» des algèbres de Banach commutatives de type fini pour construire les espaces analytiques. Cependant, les constructions de ces spectres diffèrent suivant les choix de différentes théories. Comparée à l'approche des espaces rigides à la Tate, qui est basée sur la théorie de topos, celle de Berkovich repose sur la théorie de topologie classique. En outre, les espaces analytiques de Berkovich ont souvent de bonnes propriétés topologiques. Par exemple, en général les espaces analytiques que l'on s'intéresse sont séparés. Au contraire, les espaces adiques à la Huber sont rarement séparés. De plus, pour tout schéma de type fini sur un corps non archimédien, qui est séparé (resp. connexe, propre), l'espace analytique de Berkovich associé au schéma est séparé (resp. connexe par arc, compact). Ainsi le cadre de Berkovich ressemble davantage à celui des espaces analytiques complexes. On peut espérer d'inspirer plus facilement (malgré des subtilités) les idées et résultats de la géométrie analytique complexe dans l'étude des espaces analytiques de Berkovich.

La géométrie analytique à la Berkovich a de nombreuse applications dans des contextes variés, comme par exemple géométrie d'Arakelov, dégénérescence d'une famille de variété complexe sur une base unidimensionnelle, étude de l'espace de module de courbes algébriques par la géométrie tropicale, système dynamique holomorphe, etc. Dans ces applications, on adapte souvent des résultats de la géométrie analytique complexe dans le cadre non archimédien. Cependant, cette adaptation n'est guerre automatique. En effet, bien que la théorie de formes différentielles et de courants a été introduite dans [START_REF] Chambert-Loir | Formes différentielles réelles et courants sur les espaces de Berkovich[END_REF] et [START_REF] Gubler | Positivity properties of metrics and deltaforms[END_REF] pour les espaces analytiques de Berkovich, le calcul différentiel sur les espaces de Berkovich est assez différent de celui sur les variétés analytiques complexe. En particulier, certains outils que l'on utilise couramment en géométrie analytique complexe, comme par exemple la méthode L 2 de Hörmander [START_REF] Hörmander | An introduction to complex analysis in several variables[END_REF], n'ont pas encore d'analogue adéquate dans le cadre non archimédien. L'unification de méthode dans l'étude des variétés analytiques complexes et non archimédiennes est un challenge technique ayant des applications potentielles dans divers domaines mathématiques, et surtout, du point de vue conceptuel, peut donne une explication à l'analogie entre ces deux types de variétés analytiques, malgré leurs différences en topologie et en calcul différentiel. Cette thèse contribue à une tentative dans cette direction, qui repose sur l'analyse fonctionnelle de l'algèbre des sections des puissances d'un fibré en droites ample métrisé. Le résultat obtenu peut être appliqué au problème d'extension avec un contrôle de norme d'une section le long d'une sous-variété fermé d'un fibré en droites métrisé. 0.1.2. Thème central: Fibré en droites semipositif. -Dans cette thèse, on étudie les fibrés en droites semi-positivement métrisés dans le cadre de la géométrie analytique non archimédienne à la Berkovich. Soient k un corps commutatif muni d'une valeur absolue non archimédienne, qui est complet par rapport à la topologie définie par la valeur absolue. Étant donné un schéma intègre et projectif X sur Spec k et un O X -module inversible ample L , on s'intéresse à la positivité des métriques continues sur l'analytifiée de L ,ainsique les propriété du fibré en droites métrisé pL , q sous l'hypothèse de positivité.

En géométrie analytique complexe, on considère souvent la positivité du courant de courbure d'un fibré en droites métrisé. L'avantage de cette approche est que le courant de courbure peut être calculé localement par rapport à la topologie analytique. Cette condition locale de positivité en géométrie complexe est centrale pour divers études du fibré en droites métrisé pL , q. Ces études se regroupent en deux directions thématiques. La première s'agit de l'étude de la géométrie algébrique d'une variété projective complexe en exploitant la structure de la variété différentielle sous-jacente, où la métrique joue un rôle auxiliaire. Plus précisément, la positivité portée par est combinée avec la technique de Bochner en géométrie différentielle, pour montrer des théorèmes d'annulation de groupe de cohomologie de faisceaux cohérents, ou théorème d'extension de sections avec un contrôle de norme, etc. La deuxième direction concerne l'étude des métriques eux-mêmes, comme par exemple le problème de métrique riemannienne canonique et la théorie pluripotentielle en variété analytique complexe.

Par la nature locale de cette définition de positivité pour le fibré métrisé pL , q,i l n'est pas surprenant qu'une bonne partie de démonstrations de ces résultats portent sur l'utilisation des outils d'analyse «fines», comme par exemple l'identité de Bochner-Kodaira-Nakano pour le tenseur de courbure de , la méthode de L 2 pour l'opérateur B à la Andreotti-Hörmander avec poids , ou l'analyse d'EDP non-linéaire pour l'opérateur Monge-Ampère appliqué à (pour une introduction à ces techniques, voir par exemple [START_REF] Demailly | Complex analytic and differential geometry[END_REF]).

Sous des conditions de régularité, la positivité du courant de courbure implique que la métrique s'écrit comme la limite uniforme d'une suite de métriques de Fubini-Study, construite à partir du système linéaire gradué total du fibré inversible ample L ( [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF]Theorem 3.5]). Ces conditions (locale et globale) ont des analogues dans la géométrie analytique non archimédienne. Historiquement, l'évolution de la notion de positivité pour les fibrés en droites métrisés non archimédien est inverse à celle du cas complexe, au sens où l'aspect global de la positivité a été d'abord étudié. Une notion de semipositivité pour la métrique est définie dans [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF] à l'aide de la positivité algébro-géométrique ("nef") sur les modèles entiers pour le couple pX, L q, et dans ce même article on dit qu'une métrique est "semi-ample" si elle est une limite uniforme de métriques Fubini-Study (voir aussi [START_REF] Gubler | Local heights of subvarieties over non-Archimedean fields[END_REF] et [START_REF] Chambert-Loir | Mesures et équidistribution sur les espaces de Berkovich[END_REF] pour la notion de semipositivité). Ces deux conditions de positivité sont toutes décrites par des objects globales, et ils sont finalement montrées d'être équivalentes dans [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF] [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF]. La deuxième s'affranchit l'utilisation de modèles entiers et donc plus intrinsèque à la géométrie non archimédienne, c'est pour cette raison que l'on travaille avec la definition de semipositivité par la deuxième condition suivant [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF].

Quant à une version locale pour la positivité, il faut d'abord construire une structure différentielle associée à l'espace analytique non archimédien. Un prélude apparaissait dans [START_REF] Gubler | Tropical varieties for non-Archimedean analytic spaces[END_REF], qui découvre une géométrie tropicale au sein d'un espace analytique à la Berkovich. Un théorie complète sur les formes et courants sur un espace analytique à la Berkovich est achevée dans [START_REF] Chambert-Loir | Formes différentielles réelles et courants sur les espaces de Berkovich[END_REF]( v o i ra u s s i [START_REF] Gubler | Positivity properties of metrics and deltaforms[END_REF]). Avec cette structure différentielle, il est possible de définir la positivité d'un fibré métrisé pL , q de manière locale par la positivité de son courant de courbure, exactement comme dans le cas complexe. Pourtant, la relation entre cette positivité locale avec la version globale reste encore opaque. Plus sérieusement, à ce jour, on ne dispose pas encore de tous les outils d'analyse «fine» mentionés plus haut dans la théorie développé par [START_REF] Chambert-Loir | Formes différentielles réelles et courants sur les espaces de Berkovich[END_REF]. Un obstacle clé est que l'opérateur B dans cette théorie de formes différentielle sur l'espace analytique non archimédien ne caractérise pas les objects holomorphes. Faute de cette liaison directe avec l'objet holomorphe, l'utilisation de cet opérateur ne peut pas être une transcription de celle du cas complexe.

Malgré ces subtilités dans la comparaison entre les géométries analytiques complexe et non archimédienne, nous arrivons à montrer que les fibrés en droites métrisés en géométrie analytique non archimédienne ont des propriétés analogues à celles de fibrés en droites métrisés en géométrie complexe, comme par exemple extension de section avec un contrôle de norme, ainsi que quelques aspects de la théorie pluripotentielle. Ces résultats sont résumés dans §0.2. 0.1.3. Stratégie: positivité métrique en termes de convexité holomorphe. -On étudie un fibré en droites métrisé non archimédien sous l'hypothèse de positivité globale. Comme mentionnés plus haut, une métrique semipositive au sens global peut être approchée uniformément par des métriques de modèles entiers [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF][36] ou de manière équivalente, par des métriques de Fubini-Study générales [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF]. En effet, ces deux choix correspondent à deux approches différentes pour l'étude d'un objet métrique sur un espace analytique non archimédien, dites «de Zariski» et «de Berkovich» suivant la terminologie de [14, Introduction, Table 1]. Grosso modo, le point de vue «Zariski» exploite les modèles entiers pour cet espace et utilise les techniques de la géométrie algébrique sur ces modèles, tandis que le point de vue «Berkovich» travaille sur le complété de cet espace (i.e. l'espace analytique de Berkovich associé) avec l'outil d'analyse. L'étude de fibré en droites métrisé semi-positif a déjà connu beaucoup de progrès avec une combinaison de deux approches dans divers aspects, que l'on rappellera dans §0.2. Grace aux divers résultats bien développés en géométrie algébrique, ces progrès exploitent plutôt des techniques de nature «Zariski».

Dans cette thèse on examine la positivité globale d'un fibré en droites avec le point de vue purement «Berkovich». On verra dans la section 0.2 que ceci non seulement fournit un regard intrinsèque analytique de ces aspects, mais aussi donne l'espoir d'obtenir certaines résultats sous l'hypothèse plus générale. Par exemple, dans §0.2.3, on peut espérer traiter le problème de continuité de la métrique enveloppante dans le cas de caractéristiques mixtes avec technique de nature «Berkovich», tandis que à ce jour, les techniques de nature «Zariski» ne permettent traiter que les cas de caractéristiques égales (nulle ou positive).

Pour contourner la difficulté de manque d'une structure différentielle et d'outil d'analyse «fine» en géométrie non archimédienne, l'idée adoptée dans cette thèse est de faire la géométrie analytique de manière «synthétique» et d'utiliser systématiquement des outils d'analyse «grossière », notamment l'outil d'analyse fonctionnelle. En d'autres termes, on préfère étudier directement les aspects normés des objets holomorphes, notamment les fonctions analytiques ou les faisceaux cohérents, sans faisant appel aux objets auxilliares de nature différentielle comme la courbure. Avec cette démarche, il est possible de remplacer les techniques d'analyse différentielle par les techniques d'analyse fonctionnelle. Autrement dit, on va étudier l'objet «dur» (i.e. holomorphe) avec l'analyse «grossière», au lieu d'exploiter l'objet «souple» (i.e. réel lisse) avec l'analyse «fine».

En géométrie complexe, ce point de vue pour examiner les objets holomorphes normés remonte à l'époque fondatrice de la théorie d'espace analytique, par exemple dans l'étude de faisceaux analytiques par Cartan-Serre sur la finitude de cohomologies [START_REF] Cartan | Un théorème de finitude concernant les variétés analytiques compactes[END_REF] et par Grauert sur la cohérence d'image directe [START_REF] Grauert | Coherent analytic sheaves.V o l .2 6[END_REF]. Mais il a été légèrement marginalisé après l'invention des outils puissants d'équations différentielles partielles. En géométrie non-archimédienne (théorie de Berkovich), certaines travaux de Ducros et de Poineau reprennent les outils d'analyse fonctionnelle (voir par exemple [START_REF] Ducros | Cohomological finiteness of proper morphisms in algebraic geometry: a purely transcendental proof, without projective tools[END_REF][50] [START_REF] Maculan | Notions of Stein spaces in non-archimedean geometry[END_REF]). Il se trouve que ce point de vue est parfaitement adapté à la géométrie analytique non archimédienne. On espère combiner la structure de schéma sur l'espace analytique et des outils d'analyse fonctionnelle -toutes deux bien développées dans le cadre non archimédien -pour obtenir des propriétés quantitatives analogues à celles de la géométrie complexe, parfois au prix d'optimalité.

Plus précisément, quant aux problèmes concernant un fibré en droites métrisé, on va étudier la métrique via la géométrie de son fibré en disques unités duals, tout comme l'étude d'une fonction via la géométrie de sa partie de sous-niveau. La notion clé correspondante à la positivité au sens global pour la métrique est la convexité holomorphe de son fibré en disques unités duals. L'étude de la convexité holomorphe d'une partie est bien un exemple de l'étude «d'un objet dur avec l'analyse grossière», car elle ne concerne que les fonctions holomorphes et la norme sup. C'est une démarche bien connue en géométrie complexe [START_REF] Grauert | Charakterisierung der holomorph vollständigen komplexen Räume[END_REF][10] [START_REF] Randriambololona | Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents[END_REF].

On va emprunter cette stratégie dans cette thèse pour obtenir des propriétés importantes concernant la positivité métrique d'un fibré en droites. On donne la construction du fibré en disques unités duals en termes de l'algèbre de section normée, on rappelle et développe certains aspects concernant la convexité holomorphe dans l'espace analytique non archimédien, et on obtient finalement ces propriétés en géométrie non archimédienne en passant par des arguments avec cette notion.

Résultats obtenus

Cette mémoire de thèse consiste d'un chapitre de préliminaire (chapitre 1) et trois chapitres de résultats (chapitres 2-4), dont le premier (chapitre 2) donne la construction de base géométrique, le suivant (chapitre 3) traitent le problème d'extension métrique, et le dernier (chapitres 4) étudient la métrique enveloppante. 0.2.1. Chapitre 2. -Algèbre de section normée et son spectre 0.2.1.1. Algèbre de section normée. -On propose un cadre commode pour étudier un fibré en droites métrisé semipositif pL , q non archimédien, en imitant notamment de constructions en géométrie analytique complexe dans [START_REF] Grauert | Charakterisierung der holomorph vollständigen komplexen Räume[END_REF]( v o i ra u s s i [START_REF] Bost | Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems[END_REF]Appendix]). Ce cadre nous permet de formaliser la construction de métrique Fubini-Study par des normes quotients, d'encoder la limite de ces métrique par une norme d'algèbre, et donc finalement de présenter la positivité globale comme la convexité holomorphe du spectre d'une algèbre de Banach associée.

Pour tout fibré en droites L , l'objet fondamental algébro-géométrique est l'algèbre graduée de ses sections R ' pL q :" à nPN R n pL q" à nPN H 0 pX, L bn q.

La donnée métrique est encodée en général par une norme graduée d'algèbre ~¨~(i.e. une norme graduée sous-multiplicative) sur l'algèbre de sections (voir plus bas pour les constructions naturelles d'une telle norme d'algèbre à partir d'une métrique ) @s "ps n q nPN P R ' pL q, ~s~"sup nPN ks n k n , ks n 1 `n2 k n 1 `n2 § ks n 1 k n 1 ¨ks n 2 k n 2 .

L'objet central d'étude est le séparé complété de l'algèbre de section par rapport à cette norme, qui est une algèbre de Banach commutative notée comme p R ' pL , ~¨~q. Considérons le comportement du couple pR ' pL q, ~¨~q par rapport aux morphismes de changement de base de X. Ces morphismes permettent de construire de nouvelles algèbres de section à partir de R ' pL q et de nouvelle normes d'algèbre à partir de ~¨~.P a r exemple, soit Y une sous-variété de X, l'immersion canonique de Y induit l'algèbre de sections restreintes notée comme R ' pL X|Y q, et une norme d'algèbre quotient notée comme ~¨~X |Y . On obtient alors une algèbre de Banach quotient en passant au séparé complété, notée comme p R ' pL X|Y , ~¨~X |Y q. Par ailleurs, pour tout point x P X an , l'extension de scalaires valués de k vers p pxq donne une algèbre graduée R ' pL qpxq et une norme ~¨~X |x à partir de l'ancien couple pR ' pL q, ~¨~q, par rapport à laquelle on construit le séparé complété p R ' pL qpxq. 0.2.1.2. Métriques et normes d'algèbre. -On construit quelques métriques sur L à partir de la norme d'algèbre ~¨~, tout en imitant les constructions usuelles en géométrie analytique complexe. Les immersions induites par les points de X an donnent un passage «norme vers métrique»: pour tout entier naturel n qui est suffisamment grand, la famille tk¨k n,X|x u xPX an donne une métrique continue sur L bn , notée comme FS n p~¨~q (ou simplement FSpk¨k n q). On l'appelle la n-ième métrique Fubini-Study induite par cette norme d'algèbre. La limite de la suite de métriques Fubini-Study normalisées t 1 n FSpk¨k n qu nPN est une métrique sur L , que l'on appelle la métrique enveloppante, notée comme Pp~¨~q.

Réciproquement, une métrique peut induire une norme d'algèbre, ce qui donne une passage «métrique vers norme». Plus précisément, toute métrique semi-continue supérieurement sur L induit une norme d'algèbre ~¨~ sur R ' pL q,d o n tl an-ième composant est la norme sup de n sur R n pL q, notée comme k¨k n . Ainsi, combinée avec la construction de métrique Fubini-Study plus haut, la positivité au sens global de est décrite par l'égalité de Pp q à . 0.2.1.3. Positivité globale et convexité holomorphe du spectre. -Traduisons cette description de la positivité globale de pL , q en termes de la convexité holomorphe du spectre de l'algèbre de Banach p R ' pL , ~¨~q. Le coeur est d'identifier géométriquement la métrique enveloppante Pp q à partir de cette algèbre de section normée.

On part du morphisme algébro-géométrique de contraction de section zéro dans l'espace totale VpL q du fibré dual de L , vers le cône affine CpL q de L sur X (i.e. le spectre premier de Zariski de l'algèbre R ' pL q) p : VpL qÑCpL q.

Pour étudier une métrique générale sur L , on regarde la partie compacte du fibré en disques unités duals dans VpL q an associé à pL , q, notée comme D _ pL , q.S o ni m a g e sous p an est une partie compacte dans CpL q an , qui encode toute information concernant la métrique. On examine un enrichissement métrique par la norme d'algèbre ~¨~du morphisme p, en reliant deux parties compactes contenues dans les analytifiés de ces deux espaces. D'un côté on a le fibré en disques unités duals du fibré métrisé pL , Pp~¨~qq, et de l'autre côté, on a le spectre de Gelfand d'algèbre de Banach, noté comme Mp x R ' pL , ~¨~qq. L'assertion suivante traduit la positivité d'une métrique enveloppante en termes de la convexité holomorphe d'un spectre de Gelfand d'une algèbre de Banach.

p an / / / / _ ✏ ✏ Mp x R ' pL , ~¨~qq _ ✏ ✏
VpL q an p an / / / / CpL q an Cette construction du fibré en disques unités duals donne aussi une «géométrisation» de la construction enveloppante de Pp q àp a r t i rd e . On prend la norme d'algèbre ~¨c omme celle induite par , et on note comme M ´p q la partie compacte p an pD _ pL , qq et comme Mp q la partie compacte p an pD _ pL , Pp qqq.

Corollaire 0.2.2 (Proposition 2.3.4). -Dans CpL q an , la partie compacte Mp q est l'enveloppe holomorphe de la partie compacte M ´p q. Ainsi une métrique continue est semipositive si et seulement si la partie compacte M ´p q est convexe holomorphe dans CpL q an . La construction de métrique enveloppante se comporte «fonctoriellement» par rapport aux morphismes algébro-géométriques de changement de base au sens suivant. Pour une immersion fermée Y Ñ X, on peut lui associer l'algèbre quotient R ' pL X|Y q et la norme d'algèbre quotient ~¨~ X|Y . Ce couple de quotients induit une métrique enveloppante Pp X|Y q sur L | Y . Une autre construction naturelle est de regarder l'algèbre de section Dans la pratique on s'intéresse à une version asymptotique de ce problème. Reformulé en terme de normes construites dans le chapitre 2, le problème d'extension métrique de sections restreintes devient un problème de comparaison entre deux (suites de) normes k¨k n, X|Y et k¨k n, | Y ; plus précisément, on cherche une majoration de la norme d'opérateur de l'application d'identité de pR n pL | Y q, k¨k n, | Y q dans pR n pL | Y q, k¨k n, X|Y q. Avec Théorème 0.2.3, on peut facilement déduire un résultat de l'extension métrique obtenu dans [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF]. Corollaire 0.2.4 (Theorem 3.2.1). -Soit pL , q un fibré en droites muni d'une métrique semi-positive continue. Pour tout ✏ °0, et tout t m P R m pL X|Y q il existe N P N tel que, pour tout n • N , on peut trouver

R ' pL | Y q et la
s nm P R nm pL q avec s nm | Y " t bn m et ks nm k nm § e nm✏ ¨kt m k n m | Y .
Notons que dans Corollaire 0.2.4, le seuil N dépend non-seulement des données géométriques fixés (X, Y , L , ) et de la précision d'approximation prescrite ✏,m a i s aussi du choix de la section restreinte t m . On propose deux méthodes pour améliorer ce résultat, en affranchissant la dépendance de la section restreinte. La version uniforme de l'extension métrique de sections restreintes est l'assertion suivante. Théorème 0.2.5 (Theorem 3.3.5). -Soit pL , q un fibré en droites muni d'une métrique semi-positive continue. Pour tout ✏ °0, il existe N P N tel que pour tout n • N , et tout t n P R n pL X|Y q, on peut trouver

s n P R n pL q avec s n | Y " t n et ks n k n § e n✏ ¨kt n k n | Y .
Transformons ces inégalités en deux suites de norme en une seule inégalité en deux normes d'algèbre. Si on note p✏q la métrique dilatée par e ✏ , on peut reformuler cette version uniforme en termes d'une majoration de norme d'algèbre ~¨~ X|Y par ~¨~ | Y p✏q . De manière equivalente, on cherche à majorer la norme d'opérateur de l'application identité de pR ' pL X|Y q, ~¨~ | Y p✏q q dans pR ' pL X|Y q, ~¨~ X|Y q. C'est l'énoncé suivant que l'on vise à démontrer.

Théorème 0.2.6 (Theorem 3.3.5). -Soit pL , q un fibré en droites muni d'une métrique semi-positive continue, pour tout ✏ °0, il existe C °0 tel que

~¨~ X|Y § C ¨~¨~ | Y p✏q .
Notons que ~¨~ | Y coïncide avec la norme spectrale de ~¨~ X|Y d'après Théorème 0.2.3. Ainsi, l'inégalité de cet énoncé peut être vue comme une majoration d'une norme d'algèbre par sa norme d'algèbre spectrale (avec un petit facteur de dilatation). On propose deux méthodes indépendantes pour établir cette majoration. Toute les deux sont basées essentiellement sur la propriété qu'une norme d'algèbre affinoïd est majorée par sa norme d'algèbre spectrale (sans facteur de dilatation). Pour obtenir la majoration dans l'énoncé, on fait l'approximation de notre algèbre de Banach p R ' pL , ~¨~q par les algèbres affinoïd, puis on combine les majorations particulières aux normes d'algèbre affinoïd. Un facteur de dilatation est introduit dans la majoration pour notre algèbre de Banach à cause de l'approximation. 0.2.2.2. Méthode d'approximation spectrale. -Cette méthode exploite l'approximation géométrique du spectre de p R ' pL , ~¨~q par quelques domains affinoïds. La majoration est établie en deux étapes à l'aide de techniques spectrales. On se place dans l'espace ambient CpL q an . D'abord on recouvre la partie compacte Mp p R ' pL X|Y , X|Y qq par un domain spécial (i.e. une union finie de domains affinoïds) W ✏ .O nn o t eW ✏ l'algèbre structurelle de fonction analytique et ~¨~W ✏ la norme sup sur ce domain, et c'est une algèbre de Banach par rapport à cette norme. On utilise la théorie spectrale du calcul fonctionnel holomorphe pour construire un homomorphism d'algèbres de Banach

W ✏ Ñ p R ' pL X|Y , X|Y q.
La continuité de cette homomorphisme implique la majoration de ~¨~ X|Y par ~¨~W ✏ . C'est dans cette étape où intervient essentiellement la convexité holomorphe du spectre fournie par la semi-positivité du couple pL , q. En suite, grâce au Théorème 0. Pourtant, la continuité de la métrique enveloppante est très importante dans l'étude de certains fonctionnels sur l'espace de toutes les métriques sur L , comme par exemple dans le problème de la différentiabilité du volume métrique relative. En géométrie complexe, il est connu que la continuité Pp q est impliquée par celle de , ce qui est un résultat profond concernant la régularisation globale de fonction psh [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF]( v o i ra u s s i [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]Appendix]). Sa démonstration exploite au fond les techniques L 2 pour l'opérateur B.

Pour montrer ce résultat en géométrie non archimédienne, faute d'une structure différentielle convenable et d'outil d'analyse fine, il est nécessaire de concevoir un nouveau mécanisme de régularisation pour la métrique enveloppante non archimédienne. Un tel mécanisme a été trouvé dans [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF] en termes de la notion d'idéale multiplicateur en géométrie algébrique, et par conséquence la continuité de Pp q est confirmée sous l'hypothèse de continuité de . Ils considèrent l'approximation de la métrique enveloppante par des métriques provenantes des modèles entiers pour pX, L q, suivant l'ancienne definition de semipositivité; une suite d'idéaux multiplicateur est construite pour capturer le processus d'approximation; et finalement l'effet de régularisation de ce processus provient d'une propriété noetherienne pour cette suite d'idéaux. Quelques hypothèses supplémentaires sont mises sur le corps valué de base pk, |¨|q, comme par exemple de caractéristiques égales nulle, et certaines entre elles sont enlevées dans [START_REF] Gil | Differentiability of non-archimedean volumes and nonarchimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)[END_REF] grâce à l'utilisation d'idéale test en cas de caractéristiques égales positive. Puisqu'ils ont adopté un point de vue «Zariski» pour cette problème et utilise essentiellement l'outil de la géométrie algébrique sur les modèles entiers, certaines hypothèses pour le corps de base sont indispensables dans leurs approches. VpL q an et son image M ´p q dans CpL q an . Pour un petit facteur ✏ P R °0, on considère la partie compacte M ´p q et l'inclusion dans sa dilatation le long de la direction du cône par un facteur e ✏ M ´p q ãÑ M ´p p✏qq.

Une description équivalente de la continuité de est que la dilatation soit un voisinage de cette partie compacte. On utilise cette description géométrique comme un critère pour la continuité d'une métrique. Si est continue, alors la dilatation M ´p p✏qq est un voisinage de M ´p q.O n aimerait constater que la dilatation Mp p✏qq est un voisinage de Mp q, d'où la continuité de Pp q. Notons que par Corollaire 0.2.2, la partie compacte Mp q (resp. sa dilatation) est l'enveloppe convexe holomorphe de M ´p q (resp. sa dilatation).

Cet argument indique que l'effet de régularisation provient de la «commutativité» au sens suivant. Deux opérations géométriques sur une partie compacte doivent commuter entre elles, une la formation de voisinage et l'autre la formation d'enveloppe convexe holomorphe. Dans notre exemple, ces deux opérations doivent relier M ´p q et Mp p✏qq dans le diagram commutative suivant. Les flèches horizontales signifient les opérations d'enveloppe convex holomorphe et les flèches verticales signifient les opérations de voisinage.

M ´p p✏qq

x ¨/ / Mp p✏qq an affinoid algebra norm on R ' pL X|Y q that dominates ~¨~ ,X|Y p.57 p R ' pL , q the Banach algebra of sections of L p.48 p R ' pL X|Y , X|Y q its quotient Banach algebra of restricted sections of L on Y p.48

M ´p q x ¨/ / ] O O Mp q ] O O En géométrie

VpL q

the total space of the dual of L p.32 CpL q the affine cone of L p.32 p the morphism of contraction of zero section VpL qÑCpL q p.32

D

_ pL , q the dual unit disc bundle of pL , q in VpL q an p.44 M ´p q its image under p an in CpL q an p.56 Mp q the Berkovich spectrum of p R ' pL , q in CpL q an p. In this section, one recalls some facts in functional analysis of normed vector spaces and normed algebras over a complete non-Archimedean valued field, following [START_REF] Vladimir | Spectral theory and analytic geometry over non-Archimedean fields[END_REF], [START_REF] Bosch | Non-Archimedean analysis.V o l .2 6 1 . Grundlehren der Mathematischen Wissenschaften[END_REF], [START_REF] Fresnel | Rigid analytic geometry and its applications[END_REF]and [START_REF] Temkin | Introduction to Berkovich analytic spaces[END_REF]. Besides the well-known facts, the results presented in §1.1.2, §1.2.4, §1.3.3 and §1.4 are most important for the thesis.

Throughout the section, one fixes a field k equipped with a non-Archimedean and nontrivial absolute value |¨| and we assume that k equipped with the topology defined by the absolute value is complete. Denote by k ˝the valuation ring of pk, |¨|q,b yk ˝˝the maximal ideal of k ˝,a n db y r k the residual field k ˝{k ˝˝. Denote by Hpk, |¨|q the Q-vector subspace of R generated by the set of numbers log|k ˆ|.O n e s a y s t h a t n numbers tp 1 ,...,p n u of R °0 are pk, |¨|q-free if the images of tlog p 1 ,...,log p n u in the quotient Q-vector space R{Hpk, |¨|q are Q-linearly independent therein.Unless specified, all k-algebras are supposed to be commutative and unitary (with 0 ‰ 1), and by convention all homomorphism of kalgebras are supposed to preserve the units.

1.1. Seminormed vector spaces 1.1.1. Basic constructions. -Let V be a vector space over k.W ec a l laseminorm on V any map k¨k : V Ñ R •0 which satisfies the following conditions:

(1) for any pa, sqPk ˆV , kask "|a|¨ksk,

(2) (triangle inequality) for any ps, s 1 qPV ˆV , ks `s1 k § ksk `ks 1 k.

The couple pV, k¨kq is called a seminormed vector space over k. If the following strong triangle inequality is satisfied @ps, s 1 qPV ˆV, ks `s1 k § maxtksk, ks 1 ku, we say that the seminorm k¨k is ultrametric. If ksk °0 for any non-zero vector s of V , we say that the seminorm k¨k is a norm and that pV, k¨kq is a normed vector space.

Let pV, k¨kq be a seminormed vector space over k. The seminorm k¨k induces a topology on V , a topological basis of which is given by the family of open balls ty P V | ky ´xk † ru,x P V, r °0.

Clearly k¨k : V Ñ R •0 is a continuous map with respect to this topology. Denote by npk¨kq the inverse image of t0u by k¨k. As a consequence of the triangle inequality, npk¨kq is a vector subspace of V . It is moreover closed since k¨k is continuous, and is called the null space of k¨k. Note that there exists a unique norm on V {npk¨kq, the composition of which with the projection map V Ñ V {npk¨kq identifies with the seminorm k¨k. We call this norm the quotient norm induced by the seminorm k¨k.

Let pV, k¨kq be a seminormed vector space over k.N o t et h a ti fk¨k is ultrametric, then the equality kx `yk " maxtkxk, kyku holds whenever kxk ‰ kyk.

We say that a seminormed (resp. normed) vector space pV, k¨kq is complete,o rk¨k is a complete seminorm (resp. complete norm )o nV , if any Cauchy sequence in V with respect to the seminorm k¨k admits a limit. A complete normed vector space over k is called a Banach space over k. Any finite-dimensional normed space pV, k¨kq is complete ([16, 1.2.3 Theorem 2]).

Let pV, k¨k V q be a seminormed vector space over k. Let r V c be the vector space of all Cauchy sequences in V with respect to k¨k V . We define a seminorm k¨k c on r V c which sends any Cauchy sequence tv i u iPN to lim iÑ`8 kv i k V . Denote by V c the quotient vector space r V c {npk¨k c q. Then the vector space V c equipped with the norm induced by k¨k c forms a Banach space over k, called the separated completion of pV, k¨k V q. Tautologically it can be shown that this Banach space is canonically isomorphic to the completion of V {npk¨k V q equipped with the quotient norm induced by the seminorm k¨k V . Definition 1.1.1.-Let k¨k 1 and k¨k 2 be seminorms on V .W es a yt h a tk¨k 1 and k¨k 2 are equivalent if there exist two constants C 1 °0 and C 2 °0 such that C 1 k¨k 1 § k¨k 2 § C 2 k¨k 1 . Note that this condition holds if and only if the seminorms k¨k 1 and k¨k 2 induce the same topology on the vector space V ([16, Corollaire I.3.3.1])(note that the absolute value |¨| is supposed to be non-trivial).

Definition 1.1.2.-Let pV, k¨kq be a seminormed vector space over k.I fW is a vector subspace of V , then map px P W q fi Ñ kxk defines a seminorm on W , called the restriction of k¨k on W .I fQ is a quotient vector space of V and ⇡ : V Ñ Q is the quotient map, then the map pq P Qq fi Ñ inf xP⇡ ´1ptquq kxk defines a seminorm on Q, called the quotient of k¨k on Q. Definition 1.1.3.-Let pV, k¨k V q and pW, k¨k W q be seminormed vector spaces over k, and f : V Ñ W be a k-linear map. We say that f is bounded if there exists a constant C °0 such that kf pxqk W § Ckxk V for any x P V . Note that this condition holds if and only if f is continuous with respect to the topologies on V and W induced by the seminorms k¨k V and k¨k W respectively. If on the image of f , the quotient norm of k¨k V is equivalent to the subspace norm of k¨k W , then f is said to be admissible.

We recall below several fundamental results in functional analysis and refer to [START_REF] Bourbaki | Espaces vectoriels topologiques. Chapitres 1 à 5. New. Éléments de mathématique[END_REF] Theorem 1.1.4.-Let pV, k¨k V q and pW, k¨k W q be Banach spaces over k,a n df : V Ñ W be a k-linear map. Suppose that the absolute value |¨| on k is non-trivial. Then (1) The k-linear map f is bounded if and only if its graph in V ˆW is closed under the product topology.

(2) Assume that f is bounded and surjective, then f is an open map. In particular, the quotient norm of k¨k V on W is equivalent to k¨k W .

Theorem 1.1.5.-Let V be a vector space over k and k¨k 1 and k¨k 2 be complete norms on V .S u p p o s et h a tt h ea b s o l u t ev a l u e|¨| on k is non-trivial. If there exists C °0 such that k¨k 2 § Ck¨k 1 , then the norms k¨k 1 and k¨k 2 are equivalent.

Using this norm equivalence theorem for Banach spaces over k, we have immediately the following.

Corollary 1.1.6.-Let pV, k¨k V q and pW, k¨k W q be Banach spaces over k,a n df : V Ñ W be a bounded k-linear map with closed image. Then f is admissible. Definition 1.1.7.-Let pV, k¨kq be a finite-dimensional normed vector space. The dual norm of k¨k _ on the dual vector space V _ :" Hom k pV, kq is defined by -Let pV, k¨kq be a normed vector space. Let pk 1 , |¨| 1 q be a complete valued field extension of pk, |¨|q. Set V k 1 to be V b k k 1 , which can be identified with Hom k pHom k pV, kq,k 1 q. The norm

@`P V _ , k`k _ :" sup vPV zt0u |`pvq| kvk . Remark 1.
@v 1 P V k 1 , kv 1 k k 1 :" sup ! |p`b 1qpv 1 q| 1 k`k _ , `P V _ zt0u
) defined via this identification is called the scalar extension of k¨k. -Let A be a k-algebra (the unit of which is denoted by 1)a n dk¨k be a seminorm on A (viewed as a vector space over k).

(1) The seminorm k¨k is said to be sub-multiplicative if for any pa, bqPA ˆA one has kabk § kak ¨kbk.

(2) The seminorm k¨k is called power-multiplicative if ka n k " kak n for any a P A and any n P Nzt0u.

(3) The seminorm k¨k is called multiplicative if kabk " kak ¨kbk for any pa, bqPA 2 .

A k-algebra seminorm (resp. k-algebra norm)o nA is defined to be a sub-multiplicative seminorm (resp. sub-multiplicative norm) k¨k on A such that k1k " 1. We denote by ~¨ã n algebra seminorm. Any k-algebra equipped with a complete k-algebra norm is called a Banach k-algebra.

We use letters in calligraphic style to denote Banach algebras and Banach modules (defined below) and use the same letters in literary style to denote the underlying kalgebra or the underlying module of a k-algebra. For example, a Banach k-algebra pA, ~¨~q is denoted by A.I fA 1 is a sub-k-algebra of A, then the restriction of ~¨~on A 1 is a kalgebra norm. If this norm is complete, we say that A 1 (A 1 equipped with the restricted norm) is a Banach k-sub-algebra of A.S i m i l a r l y ,i fQ is a quotient k-algebra of A, then the quotient of the norm ~¨~on Q is a sub-multiplicative seminorm. If it is a complete norm, we say that Q (Q equipped with the quotient norm) is a Banach quotient k-algebra of A.

Example 1.2.2.-Let A be a Banach k-algebra. The Tate k-Banach algebra over A of multiradius r r r "p r 1 ,...,r n qPp R °0q N is defined as the following algebra over k (for J "pj 1 ,...,j n qPN n , we denote ± iPt1,...,nu T j i i by T T T J and ± iPt1,...,nu r j i i by r r r J ) ÿ JPN n a J T T T J ,a J P A and lim |J|Ñ8 ~aJ ~¨r r r J " 0 ( with a complete k-algebra norm defined by

⌫ ⌫ ÿ JPN n a J T T T J ⌫ ⌫ T A pr r rq :" sup J ~aJ ~¨r r r J
This Banach algebra is denoted by Atr ´1 1 T 1 ,...,r ´1 n T n u, and is called an Tate A-algebra of multiradius r r r. Definition 1.2.3.-Let A 1 , A 2 be two Banach k-algebras, and : A 1 Ñ A 2 be a homomorphism of k-algebras. We say that is a homomorphism of Banach k-algebras if it is bounded as a k-linear map. A homomorphism of Banach k-algebra is often denoted by : A 1 Ñ A 2 . A homomorphism of Banach k-algebra is called an isomorphism of Banach k-algebras if there exists a homomorphism of Banach k-algebras : A 2 Ñ A 1 such that ˝ " Id A 2 and ˝ " Id A 1 . Note that we only require to be a topological homeomorphism but not necessarily an isometry.

1.2.2. Spectrum. -Let A "pA, ~¨~q be a Banach k-algebra. Let ~¨~1 be a k-algebra seminorm on A.O n es a y st h a t~¨~1 is bounded (with respect to A) if there exists C °0 such that ~¨~1 § C~¨~. Its null-space is a closed ideal I of A; the quotient k-algebra norm of ~¨~1 on the quotient k-algebra A{I is bounded with respect to the quotient k-algebra norm of ~¨~([3, Remark 1.2.2.i]). Definition 1.2.4.-Let A be a k-Banach algebra. The Berkovich spectrum MpAq is the following topological space. As a set, MpAq is defined as the set of all multiplicative seminorms on A which are bounded with respect to A.I fz is an element of MpAq,w e use the notation |¨| z to denote the corresponding seminorm on A. For any element f of A,w ed e n o t eb y|f | : MpAqÑR •0 the function which sends z P MpAq to |f | z .W e equip the set MpAq with the Berkovich topology, namely the most coarse topology which makes all functions |f | continuous, where f P A. For any subset V of MpAq, we denote by Int t pV q the topological interior of V . This topological interior is to be compared with the notion of interior of an affinoid subdomain in an affinoid domain, where the notation Int is adpoted (see [START_REF] Vladimir | Spectral theory and analytic geometry over non-Archimedean fields[END_REF]Definition 2.5.7]). For any p oint z P MpAq, let p z be the closed ideal np|¨| z q of A, which is a prime ideal. The residual field at z is defined to be the fraction field of A{p z , denoted by pzq. The quotient ring A{p z is equipped with a quotient norm of |¨| z , which is still multiplicative and hence induces an absolute value on pzq extending |¨| on k, which we also denote by |¨| z by abuse of notation. The completed residual field at z is defined to be the completion of pzq with respect to this quotient norm |¨| z , denoted as p pzq. The canonical homomorphism of k-algebras from A to pp pzq, |¨| z q is denoted by z . It is a homomorphism of Banach k-algebras. Proposition 1.2.7.-The set of points of MpAq is in canonical bijection with the set of pairs pp, |¨| 1 q where p P SpecpAq and |¨| 1 is an absolute value on the residue field at p, which is bounded from above by the quotient algebra norm induced by : A Ñ A{p on A{p.

Proof.-Apoin tz of MpAq corresponds to such a pair pp pzq, |¨| z q. Conversely, for such a pair pp, |¨| 1 q, the seminorm on A given by the composition | p¨q| 1 is obviously multiplicative and bounded from above by ~¨~. These canonical maps are inverse to each other. Let z P MpA 1 q which is not in the image of ‹ , then kerp q Ü p z ; otherwise the character A 1 { kerp qÑpzq extends to a character A 2 Ñ pzq by the density of image of , which contradicts z R Imp ‹ q. Now there exists f P kerp qzp z ,s o|f | z ‰ 0. For small enough ✏ °0, the basic open set U pf ; |f | z ´✏, |f | z `✏q Ä MpA 1 q is a neighbourhood of z which is not contained in the image of ‹ .S ot h ei m a g eo f ‹ is a closed subset in MpA 1 q.

1.2.4. Spectral seminorm. -Definition 1.2.12.-Let A be a k-algebra and ~¨~be a k-algebra seminorm on A.W e define a map ~¨~s p : A Ñ R •0 by

@f P A, ~f ~sp :" lim nÑ8 ~f n ~1 n .
Note that ~¨~s p is a k-algebra seminorm on A. In particular, the triangle inequality for ~¨~s p follows from the sub-multiplicativity of ~¨~. We call ~¨~s p the spectral algebra seminorm of ~¨~. In general, ~¨~s p is only a seminorm even if ~¨~is a norm. Remark 1.2.13.-The existence of limit is guaranteed by the (multiplicative) Fekete lemma for the sub-multiplicative sequence t~f n ~unPN . The spectral seminorm is submultiplicative, and is bounded from above by the original seminorm ~¨~. Moreover, it is power-multiplicative by construction. f n " 0 ( .

A Banach k-algebra with radical equal to t0u is said to be semi-simple. Elements in the radical are said to be quasi-nilpotent (or topological nilpotent).

Remark 1.2.16.-The radical of A contains the nil-radical of A; in other words, nilpotent elements are quasi-nilpotent. If A is semi-simple, then A is reduced. The converse may not be true.

Let A "pA, ~¨~q be a k-Banach algebra. The spectral seminorm ~¨~s p defines a quotient norm on the quotient k-algebra A{ radpAq, which is still denoted by ~¨~s p . The quotient norm ~¨~s p is bounded from above by the quotient norm of ~¨~. The uniformization A u of A is defined to be the Banach k-algebra of separated completion of pA{ radpAq, ~¨~s p q.

Conversely, if ~¨~is a power-multiplicative Banach algebra norm on A with radical t0u, then it is said to be uniform.

Obviously, ~¨~s p is bounded from above by ~¨~. It is important to note that in general ~¨~may not be bounded from above by ~¨~s p . In other words, ~¨~s p may not be complete on A{radpAq, hence the two norms may not be equivalent on this reduced algebra. (One says that A is a Banach function algebra if the two norms are indeed equivalent [8, Definition 3.8.3.1].) In general, one still has the following statement

Proposition 1.2.17.-MpAq is canonically homeomorphic to MpA u q.( [ 3, Corollary 1.3.3, 1.3.4])
1.2.5. Banach module. -One can also consider seminorms on modules over Banach algebra. Let A be a Banach k-algebra. A (semi)normed A-module is defined to be an A-module M with a (semi)norm k¨k such that pM, k¨kq is a (semi)normed vector space over k (denoted by M), and that the multiplication is bounded, in the sense that there exists C °0 such that @a P A, @m P M, ka ¨mk § C~a~¨kmk One calls a Banach A-module a normed A-module pM, k¨kq whose norm is complete.

Let

M 1 "p M 1 , k¨k 1 q, M 2 "p M 2 , k¨k 2 q be Banach A-modules and : M 1 Ñ M 2 be a homomorphism of A-modules. It is called bounded if there exists C °0 such that k pm 1 qk 2 § Ckm 1 k 1 for any m 1 P M 1 .
In this case is said to be a homomorphism of Banach A-modules, and is denoted by :

M 1 Ñ M 2 .

I na d d i t i o n ,t h eh o m o m o r p h i s m

of Banach A-modules is called admissible if it is admissible as linear map between normed-vector spaces over k (see Definition 1.1.3). Definition 1.2.18.-Let M be a Banach A-module. It is called a Banach finite Amodule if there exists l P N `and a surjective homomorphism of Banach A-modules A 'l Ñ M where A 'l is the Banach A-module corresponding to the A-module A 'l equipped with the norm pa 1 ,...,a l q fi Ñ max~a i ~. (Note that such a homomorphism is necessarily admissible.) Proposition 1.2.19.-Let A be a Banach k-algebra and M be a Banach A-module. If A is Noetherian as a k-algebra and M is finitely generated as A-module, then any

A-sub-module of M is closed, and M is a Banach finite A-module. ([29, Lemma 1.2.3] Definition 1.2.20.-Let : A 1 Ñ A 2 be a homomorphism between Banach k-algebras. It is said to be Banach finite if A 2 is a Banach finite A 1 -module. In this case A 2 is called a Banach finite A 1 -algebra.

Affinoid algebras

Affinoid algebras are a special kind of k-Banach algebras possessing good finiteness properties. These features allows one to endow a locally ringed space structure on their Berkovich spectra, namely the affinoid spaces. As a consequence, the Banach algebra norm of an affinoid algebra is equivalent to its spectral seminorm whenever the later is actually a norm.

1.3.1. Basic constructions.
-Affinoid algebras are k-Banach algebras that are quotient algebras of Tate algebras. Among them are strict affinoid algebras which have good finiteness properties such as Noetherianity. Some good properties pass to general affinoid algebra by a technique enlarging the base valued field which makes the affinoid algebra strict.

Definition 1.3.1.-For a multi-radius r r r "pr 1 ,...,r n qPR n , the algebra

ktr ´1 1 T 1 ,...,r ´1 n T n u"tf " 8 ÿ JPN n a J T T T J : a J P k, |a J |r r r J Ñ 0 as |J|Ñ8u
is called the Tate algebra over k with multi-radius r r r. Denote it by T n pr r rq.I ti sak-Banach algebra with respect to the Gauss norm of multi-radius r r r defined by

~f ~Tnpr r rq " max J |a J |r r r J
If r is of the form p1,...,1q, the corresponding Tate algebra T n prq is also denoted by T n .

Remark 1.3.2.-This Gauss norm is obviously sub-multiplicative. It is in fact multiplicative by an argument as in the proof of Gauss Lemma.

Definition 1.3.3.-A k-Banach algebra A is called an affinoid algebra if there exists an admissible surjective homomorphism from some Tate algebra ktr r r ´1T T T u to A. The Banach algebra norm on an affinoid algebra A is called an affinoid algebra norm. If one can take r r r with r i " 1 for all i Pt1,...,nu, then A is called a strict affinoid algebra. Suppose that there exists a finitely generated k-algebra B which is dense in B, then there exists an affinoid algebra norm ~¨~3 on B such that ~¨~3 • ~¨~.

Proof.-Let tb i u iPt1,...,mu be any set of generators of B, and write B as a quotient of krT 1 ,...,T m s by the homomorphism

krT 1 ,...,T m sÑB,T i fi Ñ b i
For any multi-index J P N m denote by b b b J the monomial ± iPt1,...,mu pb i q j i . For each i P t1,...,mu, denote ~bi ~by r i and denote the multi-radius consisting of tr i u iPt1,...,mu by r r r. Consider the Tate algebra norm ~¨~T r r r on krT 1 ,...,T m s, by the ultra-metricity of ~¨~and the definition of r r r,o n eh a s

@J P N m , @f J P k, ~ÿ JPN m f J ¨b b b J ~B § ~ÿ JPN m f J ¨b b b J ~Tr r r .
One can take ~¨~3 as the quotient norm of the restriction of ~¨~T r r r on krT 1 ,...,T m s to B.

To make an affinoid algebra strict, one can enlarge the base field.

Lemma 1.3.8.-Let r r r "pr 1 ,...,r n q be a multi-radius such tr i u iPt1,...,nu are pk, |¨|q-free.

Then the k-affinoid algebra K r r r :" ktr r r ´1T T T,r r rT T T ´1u"ktr r r ´1T T T,r r rS S Su{pT

1 S 1 ´1,...,T n S n ´1q is a field. ([3, Definition 2.1.1]) Lemma 1.3.9.-Let T n pr r rq be a k-Tate alegbra. It is strict if and only if r i P a |k ˆ| for all i.( [ 3, Corollary 2.1.6])
Corollary 1.3.10.-Let T n prq be a k-Tate algebra. Let I Ñ t1,...,nu be a subset of indices such that tr i u iPI are pk, |¨|q-free and |I| is maximal for this independence property. Let r r r I "pr i 1 ,...,r i 1 q, then K r r r I p b k T n pr r rq is a strict K r r r I -Tate algebra.

Corollary 1.3.11.-For any k-affinoid algebra A, there exists a multi-radius r r r I " pr i q iPI such that tr i u iPI are pk, |¨|q-free and K r r r

I p b k A is a K r r r I -strict affinoid algebra. ([3, Proposition 2.1.2]) 1.3.2. Algebraic structures: Noetherianity. -Let A be a Banach k-algebra, one denotes by A ˝the k ˝-algebra tf P A |f ~A,sp § 1u,a n db yA ˝˝the ideal of A ˝consti- tuting of elements ~f ~A,sp † 1. The r k-algebra A ˝{A ˝˝is called the reduction of A.I t can be shown that Ä T n is isomorphic to r krT 1 ,...,T n s.( [ 8,P r o p o s i t i o n5 . 1 . 2 . 2 ] )
Definition 1.3.12.-An element f P T n with ~f ~Tn " 1 is said to be regular in z n of degree d if its reduction f " pz n q d `∞0 §i §d´1 c i pz n q d´i in Tn where P k ˆand c i P krz 1 ,...,z n´1 s. Proposition 1.3.13 (Weierstrass division). -Let T n be the k-Tate algebra of multiradius p1,...,1q, then (1) Let f P T n be an distinguished element in z n of degree d,a n dg P T n be any element.

Then there exists unique r P T n´1 rz n s of degree less than d in z n and q P T n such that g " q ¨f `r. Moreover ~g~T n " maxt~q~T n , ~r~T n u

(2) Let f P T n with ~f ~Tn " 1. Then there exists a k-algebra automorphism ⌧ of T n such that ⌧ pf q is regular in z n .

( -The Gauss norm on Tate algebra is equal to its spectral norm. For a general strict redueced affinoid algebra, the Banach algebra norm is equivalent to its spectral seminorm, thanks to the compatibility of Banach algebra norms with algebraic structures. One studies the spectral norm of the Tate algebra case by direct calculation. For any affinoid algebra A, the maximal spectrum, denoted by MaxpAq is the topological subspace of MpAq consisting of points corresponding to maximal ideals. The sup norm on this subspace is denoted by ~¨~A ,spM Proposition 1.3.20.-For any f P T n , there exists z P MaxpT n q such that |f | z "~f ~Tn ([8,Proposition5.1.4.3]). OnT n , the three norms are equal: ~¨~T n " ~¨~T n,sp " ~¨~T n,spM .

One then uses Noether normalization to investigate the spectral seminorm of general affinoid algebra. -Let A be an affinoid algebra. An affinoid domain is a closed subset V of MpAq, which is homeomorphic to p◆ V q ‹ pMpA V qq for some affinoid algebra A V and Banach algebra homomorphism ◆ V : A Ñ A V , and satisfies the universal mapping property: for any Banach algebra homomorphism : A Ñ C between affinoid algebras with ‹ pMpCqq Ñ V , there exists a unique Banach algebra homomorphism :

A V Ñ C with " ˝◆V Proposition 1.3.24.-Let V be an affinoid domain in MpAq. Then V is homeomorphic to MpA V q. Moreover A V is a flat A-algebra. ([3,P r o p o s i t i o n2 . 2 . 4 ] ) Proposition 1.3.25.
-Let V be an affinoid domain in MpAq, and let W be an affinoid domain in V . Then W is an affinoid domain in MpAq.

Example 1.3.26.-Given f "p f 1 ,...,f m q and g "p g 1 ,...,g n q tuples of elements of A, p "pp 1 ,...,p m qPpR °0q m and q "pq 1 ,...,q n qPpR °0q n , the closed subset

V " MpAqpp ´1f, qg ´1q :"tz P MpAq, |f i | z § p i |g j | z • q j u
is an affinoid domain. The corresponding homomorphism of affinoid algebras is

A Ñ A V " Atp ´1 1 T 1 ,...,p ´1 m T m ,q 1 S 1 ,...,q n S n u{pT i ´fi ,g j S j ´1q
Such domains are called Laurent domains.I fn " 0, they are called Weierstrass domains.

Example 1.3.27.-Given f 1 ,...,f m ,g P A without common zero in MpAq,a n dp " pp 1 ,...,p m qPpR °0q m , the closed subset

V " MpAqpp ´1 f g q :"tz P MpAq, |f i | z § p i |g| z u
is an affinoid domain with its affinoid algebra given by

A V " Atp ´1 1 T 1 ,...,p ´1 m T m u{pgT i ´fi q
V is called a rational domain. -Let V be an admissible covering of MpAq by affinoid domains tV i u iPI , where I is a finite set. Then for a Banach finite A-module M,t h eČech complex of M with respect to V i is defined to be the complex of Banach A-modules, equipped with the usual Čech differentials

C ' pM, Vq :0 Ñ M d 0 › Ñ π JPI,|J|"1 M J d 1 › Ñ π JPI,|J|"2 M J d 2 › Ñ ..., M J :" x â iPJ A i p b A M d l : π |J|"l a J fi Ñ π |J 1 |"l`1 b J 1 ,b J 1 " ÿ iPI,J 1 "JYtiu p´1q i a J | V i
The exactness of the complex C ' pM, Vq allows for a standard construction of a structural sheaf on MpAq G . Definition 1.3.36.-Let V be any special domain in MpAq. Fix a way of writing V as î iPI V i where I is a finite set and V i " MpA V i q are affinoid algebras, one associated to

V the k-algebra A V :" kerp π iPI A V i Ñ π i,jPI A V i XV j q,
which is a Banach algebra equipped with the sup norm on V .

O MpAq pU q :" lim -› V ÄU,V special domain A V
It is also a sheaf thanks to the compactness of special domains under canonical topology. This is called the structural sheaf of MpAq. -The base valued field change of the affinoid space X is the k 1affinoid space MpA 1 q, denoted by X 1 . The base change map is defined as canonical homomorphism of topological spaces

X 1 Ñ X induced by i k 1 {k is denote by ⇡ k 1 {k . Proposition 1.3.41.-The map ⇡ k 1 {k is surjective.
Proof.-The inverse image of x P X is the image of Mpp pxq p bAq in X 1 , which is a non-empty subset by Proposition 1.2.6.

Spectral calculus

Gelfand-Shilov theory allows one to do multi-variable spectral calculus for (commutative) Banach algebras over C. In particular, one can localize a homomorphism between Banach algebras onto a neighbourhood of its spectrum. Similar theory, as developed in [3, Chapter 7], exists in the non-Archimedean base field setting.

1.4.1. Holomorphic convex envelope. -The holomorphic convexity of spectrum of a homomorphism of Banach k-algebra depends on the denseness of its image. In case where the spectrum of a homomorphism is not holomorphic convex, one can add variables to the source algebra so that spectrum of extended homomorphism is holomorphically convex. The holomorphic convex envelope of K in MpAq is defined as the subset

p K :"tz P MpAq|@ f P A, |f | z § sup z 1 PK |f | z 1 u
The subset K is said to be holomorphically convex if p K " K.

Ò : A Ò :" Atr ´1 1 T 1 ,...,r ´1 n T n uÑB such that prp x Σ q Ñ U , where pr : MpA Ò qÑMpAq is the canonical map of projection. ([3,P r o p o s i t i o n7 . 3 . 3 ] )
1.4.2. Holomorphic functional calculus. -It is easy to localize the homomorphism to holomorphic convex neighbourhood of its spectrum. For a spectrum of homomorphism which is not holomorphically convex, one uses Proposition 1.4.6 to localize the homomorphism to any neighbourhood of it.

✓ : ΓpV, O MpAq qÑB satisfying " ✓ ˝◆V , where ◆ V : A Ñ A V " ΓpV, O MpAq q is the Banach algebra homo- morphism corresponding to the inclusion V Ñ MpAq.( [ 3, Theorem 7.3.2])

Analytification of scheme of finite type

There is a construction of Berkovich spectrum for a k-algebra similar to the one for k-Banach algebra, giving rise to analytification of k-schemes of locally finite type, as developped in [3, Section 3.4].

1.5.1. Local situation. -For affine varieties, the topological space of its analytification is defined in the same way as the spectrum of Banach algebra, except that boundedness requirement of seminorms are dropped. They enjoy similar basic properties as the spectrum of Banach algebra. Proofs are of same spirit hence are omitted. Definition 1.5.1.-Let Z " SpecpA Z q be an affine k-scheme of finite type, where A Z is a k-algebra of finite type. Its Berkovich analytification Z an is the topological space constituting of all multiplicative seminorms |¨| z on A Z as points and with the canonical topology (the weakest topology making every function

z fi Ñ |f | z continuous for any f P A Z ). ([3, Remark 3.4.2]) Proposition 1.5.2.
-There is a bijective map from the set of points of Z an to the set pairs pp, |¨| z q where p P SpecpA Z q and |¨| z is an absolute value on the residue field at p. Proposition 1.5.3.-Let : A Z Ñ A W be a homomorphism of k-algebras of finite type where Z " SpecpA Z q and W " SpecpA W q. Then there is an induced continuous map ‹ : W an Ñ Z an , which sends a multiplicative seminorm |¨| w to | p¨q| w .

Proposition 1.5.4.-If is surjective, then ‹ is injective and is a closed map; if is finite, then ‹ is surjective. ( [3,P r o p o s i t i o n3 . 4 6( 6 ) ( 7 ) ] )
Proposition 1.5.5.-Let Z " SpecpA Z q be an affine k-variety, ~¨~an algebra norm on A Z and A Z be the k-Banach algebra obtained by completing A Z with respect to ~¨~.

Then the canonical homomorphism of k-algebras from A Z to A Z induces a continuous map which embeds the Berkovich spectrum MpA Z q into Z an as a compact subspace (and is closed since Z an is Hausdorff), and the Berkovich topology coincides with the induced topology from Z an .

Proof.-For any z P MpA Z q, the multiplicative algebra seminorm (or the corresponding character) z on A Z corresponds to a unique multiplicative algebra seminorm on A Z by restriction. Since A Z is dense in A Z , the family of open sets tU pf ; p, qq,fP A Z ,p , qP Ru form a basis for topology on MpA Z q, hence the inherited topology coincides with the originial topology. So the embedding is continuous, and the image of MpA Z q is compact in Z an . Since the topology on Z an is Hausdorff,t h ei m a g eo fMpA Z q is closed.

Definition 1.5.6.-An analytic function on open set U Ñ pSpec A Z q an is a map h : U Ñ ≤ zPU pzq which is a local uniform limit of rational functions: every z P U has an open neighbourhood U 1 Ñ U such that for every ✏ °0, there exists f U 1 ,g U 1 P A Z with |h ´fU 1 g U 1 | z † ✏ and gpzq‰0 for all z P U 1 . Denote by O an pU q the k-algebra of all analytic functions on U . Definition 1.5.7.-The structural sheaf O Z an on Z an is the one assigning O an pU q to an open set U . Proposition 1.5.8.-The pair pZ an , O Z an q is a locally ringed space.

1.5.2. Global situation. -One can analytify a scheme of finite type defined over k by glueing local constructions. Definition 1.5.9.-Let X be a scheme of finite type over Spec k, and write X as î X i where X i " Spec A X i are affine charts. The Berkovich analytification of pX, O X q is defined as the locally ringed space obtained by gluing the Berkovich analytification ppX i q an , O pX i q an q of each pX i , O X i q. CHAPTER 2

NORMED SECTION ALGEBRA AND ITS SPECTRUM

In this chapter, we lay the framework of our study of positively metrized line bundle.

Our basic principle is to study the positivity properties of a metrized line bundle pL , q by globally defined objects, such as the global sections of L and the sup norm induced by . The algebraic geometry of an ample line bundle L is encoded by its ring of global sections; similarly a metric acquires positivity if it is constructed from some norms on the spaces of global sections. In practice, we gather the normed vector spaces of global sections and global norm induced by pL bn ,n q, to form a normed algebra of sections p R ' pL , q.

The fondamental point of view is to interpret geometrically the positivity of a metrized line bundle as the holomorphic convexity of its dual unit disc bundle. We show that the Berkovich spectrum of the normed algebra of sections is a holomorphically convex set in the (analytification of) the Zariski spectrum of the algebra of sections. We relate this set with the dual unit disc bundle associated with the semipositive envelope metric Pp q.

Section algebra and norms

In this section we provide the framework of our study. The first part collects basic algebro-geometric objects of study, namely the section algebra and its spectrum. The second part discusses the metrization of these algebras.

2.1.1. Section algebras. -Let k be a field. Let ⇡ : X Ñ Spec k be a scheme of finite type over Spec k.

2.1.1.1. Space of global sections. -For any invertible O X -module L and any n P N •1 , one denotes by R n pL q the k-vector space of global sections H 0 pX, L bn q, and set R 0 pL q to be k. Denote by R ' pL q the graded algebra of sections

À nPN R n pL q.I fY is a subscheme of X, then one can restrict L to Y to get an invertible O Y -module which is denoted by L | Y . As above, we use the expression R ' pL | Y q" À nPN R n pL | bn Y q to denote the graded algebra of sections associated with L | Y .
If L bn is globally generated, namely the canonical homomorphism of O X -modules ⇡ ˚pH 0 pX, L bn qq Ñ L b is surjective, the canonical homomorphism ⇡ ˚pH 0 pX, L bn qq Ñ L bn induces a morphism of k-schemes

◆ n : X ›Ñ PpH 0 pX, L bn qq 2.1.1.2. Pull back by a k-morphism. -Let f : Y Ñ X be a morphism of k-schemes.
The morphisms of O X -modules L bn Ñ f ˚f ˚pL bn q induce linear maps of k-vector spaces R n pL qÑR n pf ˚L q and a homomorphism of graded k-algebras R ' pL qÑR ' pf ˚L q (for n " 0, we consider the identity map on k). For any n P N, we denote by R n pL X|Y q the image of the canonical k-linear map

H 0 pX, L bn q ›Ñ H 0 pY, f ˚pL q bn q. Then R ' pL X|Y q :" à nPN R n pL X|Y q
(with R 0 pL X|Y q set to be k) forms a graded sub-k-algebra of R ' pf ˚pL qq.

2.1.1.3. Total space and affine cone. -One denotes by VpL q the scheme SpecpSym O X L q over Spec k,a n db y⇡ L the canonical morphism of k-schemes VpL qÑX. Let O L : X Ñ VpL q be the closed immersion which corresponds to the homomorphism of

O X -algebras Sym O X pL q" à nPN L bn ›Ñ O X
of projection to the homogeneous component of degree 0. Let CpL q be the affine cone Spec R ' pL q of L and denote by $ L : CpL qÑSpec k the canonical morphism. One denotes by 0 0 0 L :Speck Ñ CpL q the rational point of CpL q corresponding to the maximal ideal

R •1 pL q :" à nPN,n•1 R n pL q.
If U is an affine open subset of X on which L trivializes, then ⇡ ´1 L pU q identifies with Spec ´à nPN H 0 pU, L bn q ¯.

Therefore the restriction maps R n Ñ H 0 pU, L bn q induce a morphism of k-schemes from ⇡ ´1 L pU q to CpL q. The gluing of such morphisms gives a morphism of k-schemes from VpL q to CpL q, which we denote by p L . These schemes are related as follows.

Proposition 2.1.1.-Assume that X is separated and quasi-compact, and L is ample.

Then the following diagram is commutative. In general, let S ' pL q be a k-subalgebra of R ' pL q with S 0 " k. We denote by CpL S' q the k-scheme of affine cone Spec S ' pL q,a n db y0 0 0 L S' :S p e ck Ñ CpL S' q the rational point of CpL S' q corresponding to the maximal ideal S •1 pL q. The inclusion morphism of algebras S ' pL qÑR ' pL q induces a morphism of affine k-schemes CpL qÑCpL S' q which is denoted by i S' . We can construct a morphism of k-schemes from VpL q to CpL S' q, which we denote by p L S' , by gluing the following morphisms: if U is an affine open subset of Y on which L trivializes, then ⇡ ´1 L pU q identifies with Spec ´à nPN H 0 pU, L bn q ¯.

X ⇡ ✏ ✏ O L / / VpL q p L ✏ ✏ ⇡ L / / X ⇡ ✏ ✏ Spec k 0 L / / CpL q $ L / / Spec k
Therefore the restriction maps S n pL qÑH 0 pU, L bn q induce a morphism of k-schemes from ⇡ ´1 L pU q to CpL S' q. From the composition of morphisms of k-algebras S ' pL qÑR ' pL qÑ à nPN H 0 pU, L bn q, one sees easily that the composition of ⇡ L and i S' is

p L S' VpL q ⇡ L ››Ñ CpL q i S' ››Ñ CpS ' q.
In general, if R ' is a graded k-algebra and S ' is a k-subalgebra of R ' ,w es a yt h a tt h e inclusion homomorphism is a (TN)-isomorphism and that S ' is (TN)-isomorphic to R ' ([33, §2.7.2]) if there exists N P N such that for any n • N , the inclusion S n Ñ R n is an isomorphism of k-linear space.

As above, we consider the invertible

O Y module L | Y .W ed e n o t eb yVpL | Y q the scheme Spec O Y Sym L | Y and by ⇡ L | Y the canonical morphism of k-schemes VpL | Y qÑY .W e denote by CpL | Y q the affine cone Spec R ' pL | Y q and by 0 0 0 L | Y the rational point of CpL | Y q coorsponding to the maximal ideal R •1 pL | Y q. We denote by p L | Y the morphism from VpL | Y q to CpL | Y q.
Moreover, we consider the subalgebra R ' pL X|Y q of R ' pL | Y q. We denote by CpL X|Y q the affine cone Spec R ' pL X|Y q,b y0 0 0 L X|Y the rational point of CpL X|Y q coorsponding to the maximal ideal R •1 pL X|Y q. The morphism CpL | Y qÑCpL X|Y q associated with the canonical inclusion R ' pL X|Y qÑR ' pL | Y q is denoted by i L X|Y .I fL is ample and X is projective, then R ' pL X|Y q is an (TN)-isomorphic subalgebra of R ' pL | Y q. Lemma 2.1.2.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. The restriction of i S' on CpL qz0 0 0 has image in CpL S' qz0 0 0, and it is an isomorphism of schemes

i S' : CpL | Y qz0 0 0 L | Y Ñ CpL S' qz0 0 0 L S' .
In particular, assume that L is ample, one can consider the (TN)-isomorphic subalgebra

R ' pL X|Y q of R ' pL | Y q, then the morphism i L X|Y : CpL | Y qz0 0 0 L | Y Ñ CpL X|Y qz0 0 0 L X|Y is an isomorphism.
Proof.-Since the canonical inclusion is a (TN)-isomorphism, there exist N P N such that S •N pL q is equal to R •N pL q.F o ra n yf P R •1 pL q,o n eh a sf N P R •N pL q,s o f N P S •N pL q. One consider the localization morphism of k-algebras induced by the canonical inclusion

◆ S' pf q : S ' pL qrf ´N sÑR •1 pL qrf ´N s»R ' pL qrf ´1s.
This an isomorphism of k-algebras. It is injective, and is surjective: as for any a P R ' pL q, one has a ¨f N P R •N ,s oa ¨f N P S •N and a " a ¨f N ¨f ´N P S ' rf ´N s. Hence the induced morphism of schemes i S' pf q :SpecpR ' pL qrf ´1sq Ñ S ' pL qrf ´N s is an isomorphism. Moreover, let g P R •1 pL q, one sees easily that the restrictions of ◆pf q and ◆pgq on S •1 pL qrf ´N srg ´N s are the same morphism of algebras ◆pfgq : S ' pL qrf ´N srg ´N sÑR ' pL qrf ´N srg ´N s»R •1 pL qrf ´1srg ´1s, so the isomorphisms of schemes i S' pf q ( f PR •1 pL q glue together into an isomorphism of schemes

Spec R ' pL q» § f PR •1 pL q Spec R ' pL qrf ´1sÑ § f N PS •N pL q
Spec S ' pL qrf ´N s»Spec S ' pL q. In a similar way, for any subalgebra S ' pL q of R ' pL q, we denote by S ' pL qpxq the graded pxq-algebra S ' pL qb k pxq,a n db y L S' the composition morphism of greded pxq-algebras

L S' : S ' pL qpxq ›Ñ R ' pL qpxq ›Ñ Sym pxq L pxq.
In particular, for the subalgebra R ' pL X|Y q of R ' pL | Y q and a point y P Y , we denote by L X|Y pyq the corresponding morphism of pyq-algebra.

Proposition 2.1.3.-Assume that there exist N P N such that for any n • N , L bn is globally generated. If S ' pL q is a (TN)-isomorphic subalgebra of R ' pL q, then for any x P X,t h em o r p h i s mo fpxq-algebra L S' pxq is a (TN)-isomorphism. In particular, L pxq, L | Y pyq and L X|Y pyq are (TN)-isomorphic.

Proof.-If L bn is globally generated, then for any x P X,t h epxq-linear map L : R n pL qpxq ›Ñ L bn pxq is surjective. So the composed linear map L S' : S n pL qpxq ›Ñ L bn pxq is also surjective for sufficiently large n. Since L bn pxq is of rank one and S n pL qpxq is of rank at most one, this linear map is an isomorphism for sufficiently large n.. So L S' is a (TN)-isomorphism. The rest of the assertion follows from this main one.

In the rest of the thesis, we make the following assumptions for algebro-geometric data. Let X be an integral projective scheme over Spec k of pure dimension d, Y be a reduced closed sub-scheme of X with its canonical closed immersion i Y : Y Ñ X,a n dL be an ample invertible O X -module. We fix an integer M P N such that L bM is very ample and that, for any n • M , the restriction map from

R n pL q to R n pL | Y q is surjective, namely R n pL | Y q"R n pL X|Y q.
2.1.2. Global norms and metrics. -We consider additional metric structures on the algebraic objects, in order to measure their size. To study line bundles, there are basically two ways of metrization: either one puts a metric on the line bundle, measuring the length of any section at each point in a local manner, or one puts a norm on the space of global sections, measuring their sizes in a global manner on X. is also an ultrametric algebra seminorm on R ' .O n eh a s~¨~ § ~¨~`,a n d~¨~`is the largest ultrametric algebra seminorm on R ' whose restriction on each homogeneous pieces R n coincides with the restriction of ~¨~.

Proof.-It's obvious that ~¨~`is ultrametric. Let ps n q nPN and pt n q nPN be elements in R ' . Let n 1 and n 2 be the smallest integers such that

~sn 1 ~"sup nPN ~sn ~, ~tn 2 ~"sup nPN ~tn ~.
Let pr n q nPN be the product element pr n q nPN ¨pt n q nPN . Thanks to the ultrametricity of ~¨~, for any m P N,o n eh a s

~rm ~"~ÿ nPN s n ¨tm´n ~ § sup nPN ~sn ~¨~t m´n ~( § ~sn 1 ~¨~t n 2 ~" ~ps n q nPN ~`¨~pt n q nPN ~`.
From this inequality one deduces that ~¨~`is sub-multiplicative. It's obvious that ~1~`" ~1~"1. Hence ~¨~`is an ultrametric algebra seminorm. Besides, if ~¨~1 is any other ultrametric algebra seminorm on R ' whose restriction on each homogeneous pieces R n coincides with the restriction of ~¨~, then one has ~¨~1 § ~¨~`by the construction.

Definition 2.1.5.-We keep the notation of the above proposition. If the equality ~¨~" ~¨~`holds, we say that ~¨~is graded.

Definition 2.1.6.-Let R ' " À nPN R n be a graded k-algebra. For any n P N, let k¨k n be a seminorm on R n . We say that the family of seminorms pk¨k n q nPN is sub-multiplicative if they satisfy the property

@pm, nqPN 2 ,s m P R m ,s n P R n , ks m ¨sn k m`n § ks m k m ¨ks n k n ; k1k 0 " 1.
It is said to be power-multiplicative if

@ n P N,s n P R n ,lP N, ks l n k ln " ks n k l n . It is said to be multiplicative if @pm, nqPN 2 ,s m P R m ,s n P R n , ks m ¨sn k m`n " ks m k m ¨ks n k n ; k1k 0 " 1. Proposition 2.1.7.-Let R ' "
À nPN R n be a graded k-algebra and pk¨k n q nPN be a sub-multiplicative family of ultrametric seminorms on pR n q nPN . Then the map ~¨~: R ' Ñ R •0 defined by ~ps n q nPN ~"sup nPN ks n k n is the unique graded ultrametric algebra seminorm on R ' , whose restriction on R n coincides with k¨k n . It is power-multiplicative (resp. multiplicative) if and only if the norm family pk¨k n q nPN is power-multiplicative (resp. multiplicative).

Proof.-The unicity is obvious. To show the properties concerning multiplication, let s "ps n q nPN and t "pt n q nPN be two elements of R ' and r "pr n q nPN be their product. One has If the family pk¨kq nPN is power-multiplicative, for any l P N, we already know that the norm ~¨~is sub-multiplicative, so ~sl ~ § ~s~l.

Let n 0 P N be the smallest integer for which ks n 0 k n 0 " max nPN ks n k n , hence ~s~"ks n 0 k n 0 .

By the choice of n 0 , for any pj 0 ,...,j m qPN m`1 such that ÿ iPt0,...,mu j i " l, ÿ iPt0,...,mu i ¨ji " ln 0 one has π iPt0,...,mu ps i q j i ln 0 § kps n 0 q l k ln 0 and the equality holds if and only if pj 0 ,...,j m q"p 0,...,0,l,0,...,0q where l is on the n 0 -th place for the exponent of s n 0 , so by the construction of ~¨~and the fact that it is ultrametric, one gets ~sl ~• kps l q ln 0 k ln 0 • kps n0 q l k ln 0 " ks n 0 k l "~s~l, hence the equality ~sl ~"~s~l holds, and ~¨~is power-multiplicative.

If the family pk¨kq nPN is multiplicative, one already knows that ~r~ § ~s~¨~t~.

To see the reverse inequality, let i 0 and j 0 be two smallest elements of N verifying the following property

ks i 0 k i 0 " max nPN ks n k n ( , kt j 0 k j 0 " max nPN kt n k n ( .
Then by the choice of i 0 and j 0 and the fact that the family is multiplicative, for any pi, jqPN 2 satisfying i `j " i 0 `j0 ,o n eh a s

ks i ¨tj k i 0 `j0 " ks i k i ¨kt j k j § ks i 0 k i 0 ¨kt j 0 k j 0 ,
and the equality holds only if pi, jq"pi 0 ,j 0 q. Since k¨k n is ultrametric, one has

kr i 0 `j0 k i 0 `j0 " ÿ i`j"i 0 `j0 s i ¨tj i 0 `j0 " ks i 0 k i 0 ¨kt j 0 k j 0 .
By the construction, one has ~s ¨t~• kr i 0 `j0 k i 0 `j0 " ks i 0 k i 0 ¨kt j 0 k j 0 "~s~¨~t~.

So ~¨~is multiplicative.

Proposition 2.1.8.-Let S, R and T be k-algebras, and let g : R Ñ T be a surjective homomorphism of algebras, f : S Ñ R be an injective homomorphism of algebras. Let ~¨~R be an ultrametric algebra seminorm on R. Then the quotient seminorm ~¨~T on T induced by g, and the sub-seminorm ~¨~S on S induced by f is also an ultrametric algebra seminorm.

Proof.-The ultrametricity of seminorm is inherited by the quotient one or the sub one. It suffices to verify the sub-multiplicativity. The sub-seminorm is obviously submultiplicative. For the quotient seminorm, let t 1 and t 2 be elements in T . Then by the definition of quotient norm, for any ✏ P R °0 one can find s 1 and s 2 in S such that

f ps i q"t i , ~si ~S § e ✏ ~ti ~T ,i" 1, 2.
Thus one has

~t1 ¨t2 ~T § ~s1 ¨s2 ~S § ~s1 ~S ¨~s 2 ~S § e 2✏ ~t1 ~T ¨~t 2 ~T ,
let ✏ tends to 0, one gets the sub-multiplicativity of ~¨~T .

Definition 2.1.9.-We keep the notation of the above proposition. The algebra norm ~¨~defined by ~ps n q nPN ~"sup nPN ks n k n is called the orthogonal sum of the submultiplicative family pk¨k n q nPN of ultrametric norms on pR n q nPN . Corollary 2.1.10.-Let R ' " À nPN R n be a graded k-algebra and ~¨~be an ultrametric seminorm on R ' .I f ~¨~is sub-multiplicative (resp. power-multiplicative, resp. multiplicative), then also is ~¨~`. Rp~¨~q can be identified with the Banach algebra consisting of sequences ps n q nPN of ± nPN R n such that the sequence p~s n ~qnPN converges to 0, with the norm given by ~¨~.

Proof.-We shall construct a bijection which is an isomorphism of algebras. In fact, any element ps n q nPN of ± nPN R n gives rise to a sequence ps 0 ,...,s m , 0,...,0,...q,m P N in R ' which is a Cauchy sequence with respect to ~¨~if the sequence p~s n ~qnPN converges to zero thanks to the ultrametricity of ~¨~. Conversely, let ps pmq n q nPN ,m P N be a Cauchy sequence of R ' with respect to ~¨~. Since the norm ~¨~is graded, one deduces that, for any n P N, the sequence ps pmq n q mPN is a Cauchy sequence in R n , which converges to an element s p8q n in R n , and that the convergence is uniform for n P N. Then one can change the limit in the following calculation:

lim nÑ8 ks p8q n k n " lim nÑ8 lim mÑ8 ks pmq n k n " lim mÑ8 lim nÑ8 ks pmq n k n " lim mÑ8 0 " 0.
The third equality holds since s pmq is an element of R ' so it has finitely many homogeneous components s pmq n which are non-zero. It is easy to verify that this bijection gives a homomorphism of algebras.

2.1.4. Spectral seminorm and radical. -Recall that (see §1.2.12), from any algebra seminorm ~¨~on a k-algebra A, one can construct the spectral seminorm ~¨~s p on A as @a P A, ~a~s p " lim nÑ8 ~an ~1 n , it is power-multiplicative. À nPN R n be an ultrametric graded k-algebra and ~¨b e a graded algebra seminorm on R ' . Then the kernel np~¨~q and the radical radp~¨~q (see Definition 1.2.15) are both homogeneous ideals of R ' . Moreover, the quotient seminorm of ~¨~on R ' {np~¨~q is also graded.

Proof.-Let s "ps n q nPN be an element in the kernel. Then 0 "~s~"max nPN ks n k n ( , so for every n P N, s n is also in the kernel. Hence the kernel is a homogeneous ideal. Let s "p s n q nPN be an element in the radical. Let i 0 P N be the smallest integer such that s i ‰ 0.F o ra n yl P N, since ~¨~is graded, one has

~sl ~• ks l i 0 k i 0 l "~s l i 0 ~, let l tends to 8,o n eo b t a i n st h a t 0 " lim lÑ8 ~sl ~• lim lÑ8 ~sl i 0 ~• 0.
Hence s i 0 is also in the radical, and so is s ´si 0 . By induction one sees that any s n is in the radical. So the radical is a homogeneous ideal.

Furthermore, if ~¨~is graded, so is ~¨~s p . This follows from the interpretation of spectral seminorm as the supremum seminorm over the spectrum of the completion Banach algebra (Theorem 1.2.14). Proof.-The first assertion follows from Corollary 2.1.10. For the second assertion, note that there exists C °0 such that for any s "p s n q nPN , since |¨| z is bounded by ~¨~,o n e has

@n P N, |s n | z § C ¨ks n k n ,
since ~¨~is graded, one has

|s| z " max nPN |s n | z § C ¨max nPN ks n k n " C ¨~s~.
So |¨| z is also bounded from above by ~¨~, and it corresponds to a point in Mp p R ' , ~¨~q.

Proposition 2.1.15.-Let R ' " À nPN R n be a graded k-algebra and ~¨~be an ultrametric graded algebra seminorm on R ' , then its spectral seminorm ~¨~s p is also graded.

Proof.-By Corollary 2.1.10, since ~¨~s p is ultrametric and power-multiplicative, the seminorm ~¨~s p is also power-multiplicative, and one has ~¨~s p § ~¨~s p .

For the reverse inequality, for s " ∞ iPt0,...,mu s i P R ' , let n 0 P N be the smallest integer such that ~sn 0 ~sp is maximal among all t~s n ~sp u nPN . Then for any n one has ~sn ~sp § ~sn 0 ~sp and the inequality is strict for n † n 0 . Since there are only finitely many terms in s, one can find ⇢ † 1 and N P N and for any ✏ °0 a constant C ✏ P R °0, such that @i, @j i P N,

~sj i i ~ § C ✏ ¨e✏j i ~sn 0 ~ji sp ; @i † n 0 , @j i • N, ~sj i i ~ § ⇢ j i ¨~s n 0 ~ji sp .
Take l P N and consider the monomial terms in ps l q l¨n 0 . They are of form ± s j i i where ∞ i j i " l and ∞ i i ¨ji " ln 0 . By the above estimates, for one such term one has (1)

~π i s j i i ~ § π i ~sj i i ~"π iPA ~sj i i ~¨π iPB ~sj i i ~ § C |A| ✏ ¨e✏ ∞ iPA j i ¨⇢∞ iPB j i ¨~s n 0 ~l sp
where B is the subset of i in t0,...,mu such that i † n 0 and j i • N ,a n dA is its complement with |A| § m `1.W r i t ej n 0 " l ¨ for some Pr0, 1s, then one has

ÿ i †n 0 j i `ÿ i°n 0 j i " lp1 ´ q, ÿ i †n 0 i ¨ji `ÿ i°n 0 i ¨ji " lp1 ´ qn 0 .
Each summand is non-negative, thus one gets

lp1 ´ qn 0 • ÿ i°n 0 i ¨ji • pn 0 `1q ÿ i°n 0 j i • pn 0 `1qrp1 ´ ql ´ÿ i †n 0 j i s, which implies that ∞ i †n 0 j i • n 0 n 0 `1 p1
´ ql. Thus one has

∞ iPB j i • n 0 n 0 `1 p1 ´ ql ´pn 0 1q¨N . Denote by ⌧ the number n 0 n 0 `2 p1 ´ q. If j n 0 † l, namely † 1, then ⌧ ‰ 0 and for large l P N,o n eh a s ∞ iPB j i • ⌧ ¨l thus ∞ iPA j i § p1 ´⌧ q¨l.
Choose ✏ such that e ✏p1´⌧ q ¨⇢⌧ † 1, thus for large l in (1) one gets ~±i s j i i ~ † ~sn 0 ~l sp . The term where j n 0 " l is the single one s l n 0 and one has ~sl n 0 ~• ~sn 0 ~l sp . Thus by the ultrametricity one gets ~ps l q ln 0 ~"s l n 0 ~for large l and by the gradedness of ~¨~one has ~sl ~• ~ps l q ln 0 ~• ~sn 0 ~l sp , so ~s~s p • ~sn 0 ~sp . Hence ~¨~s p • ~¨~s p .F i n a l l y~¨~s p is graded. In fact, by Remark 1.3.2, the Gauss-Tate norm is multiplicative, so it is equal to its spectral norm. They are graded with respect to the grading by total degree, and even better, graded with respect to the multi-grading by the multi-degree of krT 1 ,...,T n s. The particularity of case n " 1 is that the gradings by total degree and by multi-degree coincide.

Corollary 2.1.18.-Let R ' " krT s be the one-variate polynomial algebra and S ' Ñ R ' be a (TN)-isomorphic subalgebra. Let ~¨~be an ultrametric graded powermultiplicative algebra seminorm on S ' , then one can find a unique ultrametric graded power-multiplicative algebra seminorm on R ' , denoted by ~¨~1, such that the restriction of ~¨~1 on S ' is equal to ~¨~. One can find n 0 such that S n 0 " R n 0 and define the norms as follows

kT k 1 1 :" kT n 0 k 1 n 0 n 0 , kT n k 1 n :"pkT k 1 1 q n .
Since the family k¨k n ( nPN is power-multiplicative, one can easily verify that this assignment of k¨k 1 n ( nPN does not depend on the choice of n 0 , and the new family of norms is still power-multiplicative. By Proposition 2.1.7, the orthogonal sum of this family is a power-multiplicative algebra seminorm, which we denote by ~¨~1. Its restriction on S ' is indeed ~¨~. -Let k be equipped with a complete ultrametric absolute value |¨|. Let X an the Berkovich analytic space associated with X and denote by j X : X an Ñ X the morphism of locally ringed spaces sending any x P X an to its associated scheme point (see §1.5 for a reminder on the construction of X an and j X ). Remember that any analytic point x P X an corresponds to a pair pjpxq, |¨| x q where |¨| x is an absolute value on the residue field pjpxqq. For simplicity, we shall use the notation pxq instead of pjpxqq, and we denote by p pxq the completed residue field. If L is an invertible O X -module, we denote by L an the pull-back of L by j X .

2.1.5.2. Analytic total space and affine cone. -Let L be an invertible O X -module. For any x P X an , one denotes by p pxq the completed residual field at x,andb yL pxq the rank one p pxq vector space L b O X p pxq.

Proposition 2.1.20.-One has the following commutative diagram of morphisms of analytic spaces

X an ⇡ an ✏ ✏ O an L / / VpL q an p L ✏ ✏ ⇡ an L / / X an ⇡ an ✏ ✏
pSpec kq an 0 0 0 an L / / CpL q an $ an L / / pSpec kq an Moreover, the restriction of p an L to VpL q an zO an L pXq is an open immersion, whose image is contained in CpL q an z0 0 0 an L .I fX is proper, then p L restricted to VpL q an zO an L pXq defines an isomorphism between VpL q an zO an L pXq and CpL q an z0 0 0 an L .

Proof.-This follow from Proposition 2.1.1 and Proposition 1.5.10.

Proposition 2.1.21.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. Then restriction of the morphism of k-analytic spaces Spec R ' pL q an Ñ Spec S ' pL q an on Spec R ' pL q an z0 0 0 an L has image in Spec S ' pL q an z0 0 0 an L S' , and this restriction is an isomorphism of analytic spaces Spec R ' pL q an z0 0 0 an L Ñ Spec S ' pL q an z0 0 0 an

L S'
Proof.-This follows from Proposition 2.1.2 and Proposition 1.5.10.

2.1.5.3. Analytic fiber of total space.-F o ra n yx P X an , one can consider the base change map x : Mpp pxqq Ñ X an and the corresponding fiber product VpL q an b O X p pxq which is a p pxq-analytic space given by the analytification of the algebraic spectrum of the graded p pxq-algebra Sym p pxq L pxq. The fiber product gives rise to a commutative diagram of morphism of analytic spaces

VpL q an pxq ✏ ✏ / / VpL q an ⇡ an L ✏ ✏ Mpp pxqq x / / X an .
For any x P X an and any subalgebra S ' pL q of R ' pL q we shall denote by S ' pL qpxq the graded p pxq-algebra S ' pL qb k p pxq. Let w P VpL q an be a point such that ⇡ an L pwq"x, then it determines a unique point in VpL q an pxq in the following way. One can find an affine Zariski open set U of X containing the schematic point jpxq on which L trivializes. Denote by R ' pL | U q the graded algebra à nPN H 0 pU, L bn q, then ⇡ an L is identified with Spec R ' pL | U q an . The point w corresponds to a multiplicative seminorm on the k-algebra R ' pL | U q. Since ⇡ an L pwq"x, one knows that ⇡ L pjpwqq " jpxq, and that the restriction of |¨| w on pjpwqq to pjpxqq is |¨| x . Hence w induces a unique multiplicative seminorm on the p pxq-algebra

R ' pL | U qb k p pxq"Sym p pxq L pxq.
By abuse of notation, we shall denote this point still by w. conversely, if w is a point in VpL q an pxq, it determines a unique point in VpL q an which we still denote by w,a st h e multiplicative seminorm on R ' pL | U q which comes from the following composition

R ' pL | U qÑSym p pxq L pxq |¨|w ››Ñ R 2.1.6.
Metrics. -2.1.6.1. Pseudometrics as real valued function. -Let F X be the sheaf of real-valued functions on X an . By pseudometric on L one refers to a morphism of sheaves of sets : L Ñ j X,˚p F X q such that, for any x P X an ,t h em a p|¨| pxq : L pxqÑR induced by is a seminorm on the one-dimensional vector space L pxq over p pxq. The pair pL , q is called a pseudometrized invertible O X -module. If, for any x P X an ,t h em a p|¨| pxq is a norm, one says that is a metric.I fs is a section of L on an open neighbourhood of x, then |spxq| pxq is denoted by |spxq| for simplicity.

We say that a pseudometric is upper semi-continuous (resp. continuous ) if, for any Zariski open subset U of X and any section s P ΓpU, L q, the function px P U an q ›Ñ |spxq| is upper semi-continuous (resp. continuous), or equivalently, in additive notion, the function px P U an q ›Ñ´log|spxq| is lower-semicontinuous (resp. continuous).

2.1.6.2. Pull-back by a morphism. -Let f : Y Ñ X be a morphism of separated kschemes of finite type. Let L be an invertible O X -module, equipped with a pseudometric . We define a pseudometric f ˚ on f ˚pL q such that, for any section s of L on a Zariski open subset U of X,o n eh a s @ y P f ´1pU q an , |f ˚psq| f ˚ pyq"|s| pf an pyqq.

The pseudometric f ˚ is called the pull-back of by f . Since f an : Y an Ñ X an is continuous (Proposition 1.5.10), if the metric is continuous, also is f ˚ .I f Y is a subscheme of X and if f : Y Ñ X is the canonical immersion, then the pull-back metric f ˚ is also denoted by | Y .

2.1.6.3. Metrics on the trivial line bundle.-A n ym a pf : X an Ñ R Yt`8u determines a pseudometric ⌧ f on O X such that, for any regular function a of X on a Zariski open subset U , one has (with the convention e ´8 " 0)

@ x P U an , |apxq| ⌧ f " |apxq| ¨e´fpxq .
Note that f fi Ñ ⌧ f defines a bijection between the set of maps X an Ñ R `Yt `8u and that of pseudometrics on O X , which maps the set of real-valued functions bijectively to that of pseudometrics on O X . Moreover, the pseudometric ⌧ f is continuous if and only if f is continuous on X an . The trivial invertible sheaf O X equipped with the pseudometric ⌧ f is denoted by O X pf q. The metric corresponding to the identically vanishing function is called the trivial metric on O X .

2.1.6.4. Distance between two metrics. -Let 1 and 2 be two metrics on L . The distance of these two pseudometrics is a generalized positive real number (in R `Yt`8u) defined as

distp 1 , 2 q" sup xPX an ˇˇˇl og ˇˇ 1 pxq 2 pxq ˇˇx ˇˇˇ.
If X is proper and 1 , 2 are continuous metrics, then distp 1 , 2 qPR `.

2.1.6.5. Tensor product pseudometrics. -Let L 1 and L 2 be invertible O X -modules, and 1 and 2 be pseudometrics on L 1 and L 2 respectively. The pseudometrics 1 and 2 induce by passing to tensor product a pseudometric 1 ` 2 on L 1 b L 2 such that, For any Zariski open subset U of X and any ps 1 ,s 2 qPΓpU, L 1 qˆΓpU, L 2 q,o n eh a s

@ x P U an , |ps 1 ¨s2 qpxq| 1 ` 2 " |s 1 pxq| 1 ¨|s 2 pxq| 2 .
If 1 and 2 are continuous, then 1 ` 2 is also continuous.

In particular, for any ✏ P R, we denote by p✏q the pseudometric `⌧e ✏ on L bO X -L . Moreover, any metric on L determines by passing to its dual a metric ´ on L _ such that, for any Zariski open subset U of X and any ps, ↵qPΓpU, L qˆΓpU, L _ q,o n e has @ x P U an , |↵psqpxq| " |↵pxq| ´ ¨|spxq| .

If the metric is continuous, so is ´ . Let L be an invertible O X -module, and n P Nzt0u. A pseudometric on L determines by tensor power a pseudometric on L bn for any n P Nzt0u, denoted by n . By convention, 0 denotes the trivial metric on L b0 -O X (see §2.1.6.3 above).

Similarly, assume given a pseudometric on L bn . We denote by 1 n the pseudometric on L such that, for any Zariski open subset U of X and any section s P ΓpU, L q,o n eh a s

@ x P U an , |spxq| 1 n " |s n pxq| 1{n .
If the pseudometric is continuous, then so is 1 n . 2.1.6.6. Dual disc bundle. -Consider the fiber of the canonical projection ⇡ an over x P X an , which is given by pSpec Sym p pxq L pxqq an . Any pseudometric determines a family of ultrametric seminorms |¨| n pxq ( nPN on the graded pieces L bn pxq ( nPN . This family is multiplicative, so it corresponds to a graded ultrametric multiplicative seminorm on Sym p pxq L pxq by Proposition 2.1.7, denoted by ~¨~ pxq, which is a Gauss-Tate seminorm by Proposition 2.1.16. We denote by z Sym p pxq pL pxq, q the completion p pxq-Banach algebra of Sym p pxq L pxq with respect to this seminorm. We denote by D _ pL , qpxq the p pxq-affinoid domain MpSym p pxq L pxqq which is a disc in the p pxq-analytic space Vpxq an . Remember that any w P VpL q an determines a unique point in VpL q an , still denoted by w, which is a multiplicative seminorm on Sym p pxq L pxq (see §2.1.5.3). Let be a pseudometric on L, the following topological subspace of VpL q an w P VpL q an : @e 1 P L pxq,

|e 1 | w § |e 1 | presp. † |e 1 | q (
equipped with induced topology, is called the dual closed (resp. open) disc bundle of radius 1 of the pseudometrized pair pL , q. We denote it by D _ pL , q (resp. D _ pL , q). Proposition 2.1.22.-Consider the restriction of the continuous map ⇡ an to the topological subspace D _ pL , q.F o ra n yx P X, the topological fiber of the restriction of ⇡ an coincides with the underlying topological space of D _ pL , qpxq.

Proof.-Let w P VpL q an with ⇡ an pwq"x. ( xPX an gives rise to a metric on L bn , which we call the Fubini-Study metric associated with k¨k n on R n pL q,d e n o t e db yFSpk¨k n qpxq.T h em e t r i c 1 n FSpk¨k n q on L is called the normalized Fubini-Study metric associated with k¨k n on R n pL q.

Since
In particular, if the norm k¨k n is ultrametric and admit an ultrametric orthogonal basis, namely there is a basis ts n,j u jPt1,...,dnu of R n pL q such that @p 1 ,..., dn qPk dn , k dn ÿ j"1 j ¨sn,j k n " max jPt1,...,dnu k j ¨sn,j k n , then the corresponding Fubini-Study metric FSpk¨k n q is said to be diagonalizable ( [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF]). Previously, such metrics are already studied and heavily used in [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF].

Moreover, let S ' pL q be a subalgebra of R ' pL q, equipped with a family of norms k¨k n ( nPN on W n ( nPN . If for some n P N, the evaluation map S n pL qb k p pxqÑL bn pxq is surjective, one can define the Fubini-Study metric 1 n FSpk¨k n q on L . 2.1.7.2. Fubini-Study envelope metric associated with an algebra norm. -Let L be an invertible O X -module which is generated by global sections. Let k¨k n ( nPN be a family of seminorms on R n pL q ( nPN . If the sequence of pseudometrics t 1 n FSpk¨k n qu nPN converges pointwisely to a limit pseudometric, we denote it by Pptk¨k n u nPN q and call it the Fubini-Study envelope pseudometric associated with tk¨k n u nPN .

In particular, let ~¨~be an algebra seminorm on R ' pL q, and let tk¨k n u nPN be the associated sub-multiplicative family of seminorms on tR n pL qu nPN . From these data, one can construct an envelope metric in two equivalent ways, in terms of family of norms of restriction to homogeneous components or in terms of the algebra norm itself:

-Since the family tk¨k n u nPN is sub-multiplicative, also is the family tk¨k n,p pxq u nPN .

Then the family of quotient seminorms tFSpk¨k n qpxqu nPN is also sub-multiplicative by Proposition 2.1.7 and Proposition 2.1.8. Hence the norms t 1 n FSpk¨k n qpxqu nPN converges to a limit seminorm on L pxq by Fekete Lemma. This gives rise to a pseudometric on L , called the Fubini-Study envelope pseudometric associated with ~¨~. We denote it by Pp~¨~q, thus the limit seminorm is Pp~¨~qpxq.T h eF u b i n i -Study envelope metric is not necessarily continuous.

-For any x P X an , the evaluation map induces a p pxq-algebra homomorphism where N is an integer satisfying R N pL qpxq"L bN pxq.

R ' pL
Moreover, let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. Since there exist N P N such that the evaluation maps S n Ñ L bn pxq are surjective for all n • N , one can construct in a similar way a pseudometric on L , called the Fubini-Study envelope pseudometric associated with ~¨~on S ' pL q. It is constructed either as the limit pseudometric of the sequence 1 n FSpk¨k n q

( nPN •N
, or as the spectral seminorm ~¨~1 X|x;sp on Sym p pxq L pxq as the unique graded ultrametric power-multiplicative extension of ~¨~X |x;sp on R ' pL qpxq.

2.1.7.3. Semipositive metrics. -A metric on L is said to be semipositive if it is a Fubini-Study envelope metric and the convergence is uniform for x P X an ([11, Definition 6.1]). Note that if the convergence is uniform for x P X an , since Fubini-Study metrics t 1 n FSpk¨k n qu nPN are continuous, the envelope metric will also be continuous. Conversely, suppose that X is proper over Spec k and that the envelope metric is continuous, then as X an is Hausdorff and compact by Theorem 1.5.11, the convergence is uniform in x P X an . semipositive metrics are thus continuous.

Note that the semipositive metric in this sense is equivalent to the notion of asymptotically Fubini-Study metric by the terminology of [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF]. We refer to [13, §5.4] and [11, §6.1] for a clear discussion of other various notions of semipositivity that have been proposed and studied historically in [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF], [START_REF] Gubler | Local heights of subvarieties over non-Archimedean fields[END_REF], [START_REF] Moriwaki | Free basis consisting of strictly small sections[END_REF], [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF], [START_REF] Chambert-Loir | Formes différentielles réelles et courants sur les espaces de Berkovich[END_REF], [START_REF] Ignacio | Arithmetic positivity on toric varieties[END_REF], [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF], [START_REF] Gubler | On Zhang's semipositive metrics[END_REF] and literatures therein.

2.1.8. Global norms associated with a metric. -We construct norms on spaces of global sections by taking the supremum norm of |p˚qpxq| associated with a metric on L for all x P X an . We can also construct quotient norm of sup norm on the space of restricted global sections.

2.1.8.1. Sup norm.-A s s u m et h a tX is proper over Spec k.N o t et h a tX an is then a compact Hausdorff space (see [START_REF] Vladimir | Spectral theory and analytic geometry over non-Archimedean fields[END_REF]Theorem 3.4.8]). Let L be an invertible O X -module and be an upper semicontinuous metric on L .A sX an is compact, any upper semicontinuous function on X an is bounded from above and attains its maximal value. In particular, for any n P N and any s P R n pL q,o n eh a s

ks n k n :" sup xPX an |s n pxq| † `8.
This is a norm on R n pL q. It is ultrametric since the absolute value |¨| on k is non-Archimedean and L bn is of rank 1. This gives rise to a family of norms tk¨k n u nPN . Proposition 2.1.24.-The family tk¨k n u nPN of ultrametric norms is sub-multiplicative and power-multiplicative. The orthogonal sum norm on R ' pL q, @s "ps n q nPN , ~s~ :" sup

nPN ks n k n .
is ultrametric and graded, sub-multiplicative and power-multiplicative.

Proof.-In fact, since 0 is the trivial metric on O X ,onehas~1~0 " 1, where 1 denotes the unit section of O X .F o rs n P R n pL q and s m P R m pL q,w eh a v e ks m ¨sn k n " sup |s n pxq| n q l " ks n k l n .

By Proposition 2.1.7, the ultrametric graded norm ~¨~ is sub-multiplicative and powermultiplicative. It is an algebra norm.

One denotes by p R ' pL , q the separated completion of the normed k-algebra pR ' pL q, ~¨~ q. More generally, if R ' is a graded sub-k-algebra of R ' pL q,b ya b u s e of notation we still denote by ~¨~ the restriction of the norm ~¨~ on R ' and denote by p R ' p q the separated completion of the normed algebra pR ' , ~¨~ q. The restricted norm is also power-multiplicative. In particular, for any N P Nzt0u, if one takes R ' to be À nPN R nN pL q, denoted by R pN q ' pL q, we denote by p R pN q ' pL , q the separated completion of pR pN q ' pL q, ~¨~ q.

2.1.8.2. Quotient norm.-A s s u m et h a tX is proper over Spec k. Let L be an invertible O X -module and be an upper semicontinuous metric on L . Let f : Y Ñ X be a morphism of k-schemes of finite type. Let k¨k n ,X|Y be the quotient norm of k¨k n on R n pL X|Y q. It is ultrametric and graded. Let ~¨~ ,X|Y be the quotient norm of ~¨~ on R ' pL X|Y q. In fact, @t "pt n q nPN P R ' pL X|Y q, ~t~ ,X|Y " sup

nPN kt n k n ,X|Y . Proposition 2.1.25.-~¨~ ,X|Y is sub-multiplicative. It is an algebra norm.
One denotes by p R ' pL X|Y , X|Y q the separated completion of the normed k-algebra pR ' pL X|Y q, ~¨~ X|Y q. In particular, for any N P N, we denote by R pN q ' pL X|Y q the graded k-algebra À nPN R nN pL X|Y q. This is a sub-algebra of R ' pL X|Y q. The restriction of ~¨~ ,X|Y on this sub-algebra is still denoted by ~¨~ ,X|Y . One denotes by p R pN q ' pL X|Y , X|Y q the separated completion of pR pN q ' pL X|Y q, ~¨~ ,X|Y q. In particular, if f is the canonical immersion associated with a sub-scheme, we get a Banach k-algebra p R pN q ' pL X|Y , X|Y q.

2.1.8.3. Radical of algebra norms associated with a metric. -Recall that the radical of an algebra seminorm ~¨~on A consists of the elements a P A such that ~a~s p " 0. The radical is denoted by radp~¨~q. Proposition 2.1.26.-The radicals radp~¨~ q,r a d p~¨~ ,X|Y q and radp~¨~ | Y q are all t0u. In particular, the Banach k-algebras p R ' pL , q, p R ' pL | Y , | Y q and p R ' pL X|Y , X|Y q are semi-simple (see Definition 1.2.15) and reduced.

Proof.-Since ~¨~ and ~¨~ | Y are multiplicative norms, their radicals are t0u by definition. Let t "pt n q n be an element in the radical of radp~¨~ ,X|Y q, then since ~¨~ ,X|Y is graded, the radical is also graded by Lemma 2.1.13. For every n P N, by the construction of norms associated with metric ,o n eh a s In this thesis, we make the overall assumption on the metric data, that is an uppersemicontinuous metric on L .

0 " lim lÑ8 kt l n k 1 l nl ,X|Y • lim lÑ8 kt l n k 1 l nl | Y , so t n is in the radical of ~¨~ | Y , which is t0u. Thus t is equal to 0.

Fubini-Study metrics

We give explicit expressions for Fubini-Study metrics induced by ultrametric norms admitting ultrametric orthogonal basis, and list some basic properties for operations of passage between norm and metric. Many of the results here are also obtained previously in [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF]o ri n[ 11, Section 6].

Explicite calculation for

Fubini-Study metric with orthogonal basis. -Lemma 2.2.1.-Assume that L bn is globally generated. Let k¨k n be a norm on R n pL q and let FSpk¨k n q be the associated Fubini-Study metric. Then for any x P X an and e n P L bn pxqzt0u,

|e n | FSpk¨knq " inf Pp pxq ˆ,s nPRnpL q snpxq" ¨en | | ´1 x ¨ks n k.
Proof.-This follows from Lemma 1.1.11.

If the ultrametric norm k¨k n admits an orthogonal basis, one can calculate explicitly the associated Fubini-Study metric FSpk¨k n q. Lemma 2.2.2.-Let pk 1 , |¨| 1 q be a complete ultrametric valued field extension of pk, |¨|q. Then for any tuple pr 1 ,...,r n qPR n `,o n eh a s inf p 1 ,...,nqPpk 1 q n  1 `¨¨¨`n"1 max jPt1,...,nu

| j | 1 ¨rj ( " min jPt1,...,nu tr j u.
Proof.-On the one hand, let j 0 Pt1,...,nu be an index such that r j 0 " min jPt1,...,nu r j ( .

By taking  j " 0 for j ‰ j 0 and  j 0 " 1, one sees that inf p 1 ,...,nqPpk 1 q n  1 `¨¨¨`n"1 max jPt1,...,dnu

| j | 1 ¨rj ( § r j 0 " min jPt1,...,dnu tr j u.
On the other hand, since |¨| 1 is ultrametric, if ∞ jPt1,...,nu  j " 1, then there exists at least one j 1 Pt1,...,nu such that

| j 1 | • 1,s o inf p 1 ,...,nqPpk 1 q n  1 `¨¨¨`n"1 max jPt1,...,nu ! | j | 1 ¨rj ) • inf p 1 ,...,nqPpk 1 q n  1 `¨¨¨`n"1 | j 1 | 1 ¨rj 1 • inf p 1 ,...,nqPpk 1 q n  1 `¨¨¨`n"1 r j 1 " r j 1 • min jPt1,...,nu tr j u.
Hence the two sides are equal.

Proposition 2.2.3.-Assume that L bn is globally generated. Let k¨k n be an ultrametric norm on R n pL q which admits an ultrametric orthogonal basis ts n,j u jPt1,...,dnu . Then for any x P X an and e n P L bn pxqz0, Since both these spaces are equipped with distances, we can study the quantitative features of these operations.

|e
Lemma 2.2.5.-Assume that L is globally generated. Let 1 and 2 be two uppersemicontinuous metrics on L , then

distpk¨k 1 , k¨k 2 q § distp 1 , 2 q.
(see also [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF]Lemma 6.10])

Proof.-For any s P R 1 pL q, since X an is compact, and the functions |spxq| 1 and |spxq| 2 are upper-semincontinuous, one can find a point x 1 P X an where the maximum ksk 1 of |spxq| 1 is attained, and a point x 2 P X an where the maximum ksk 2 of |spxq| 2 is attained.

If ksk 1 • ksk 2 ,o n eh a s ˇˇlog ksk 1 ksk 2 ˇˇ § ˇˇlog ˇˇs px 1 q spx 1 q ˇˇx 1 ˇˇ § distp 1 , 2 q. Otherwise ksk 1 § ksk 2 ,o n eh a s ˇˇlog ksk 2 ksk 1 ˇˇ § ˇˇlog ˇˇs px 2 q spx 2 q ˇˇx 2 ˇˇ § distp 1 , 2 q.
Hence the desired inequality holds.

Lemma 2.2.6.-Assume that L is globally generated. Let k¨k and k¨k 1 be two norms on R 1 pL q, then distpFSpk¨kq, FSpk¨k 1 qq § distpk¨k, k¨k 1 q. (see also [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF],E q u a t i o n( 6 . 2 ) ] )

Proof.-For any x P X an , let e P L pxqzt0u. Let s, s 1 P R 1 pL q and , 1 P pxq be elements such that spxq" ¨e, kek X|x " | | ´1ksk,

s 1 pxq" 1 ¨e, kek X|x " | 1 | ´1ks 1 k 1 . If kek X|x °kek 1 X|x ,o n eh a s distpkek X|x , kek 1 X|x q" ˇˇlog | | ´1ksk | 1 | ´1ks 1 k 1 ˇ § ˇˇlog | | ´1ksk | | ´1ksk 1 ˇˇ § distpk¨k, k¨k 1 q. Otherwise, one has distpkek X|x , kek 1 X|x q" ˇˇlog | | ´1ksk | 1 | ´1ks 1 k 1 ˇ § ˇˇlog | 1 | ´1ks 1 k | 1 | ´1ks 1 k 1 ˇˇ § distpk¨k, k¨k 1 q.
Varying x and taking the supremum for x P X an , one gets the desired inequality.

Lemma 2.2.7.-Assume that L is globally generated. Let k¨k 1 be an ultrametric norm on R 1 pL q, then FSpk¨k 1 q is a continuous metric on L .( [ 24,P r o p o s i t i o n3 . 1 ] )

Proof.-If k¨k 1 admit an ultrametric orthogonal basis, then from the explicite expression in Proposition 2.2.3, it's easy to see that FSpk¨k 1 q is continuous. In general, by Proposition 1.1.17, for any ✏ °0, one can approximate k¨k 1 by a ultrametric norm k¨k 1 1 admitting orthogonal basis, such that distpk¨k 1 , k¨k 1 1 q § ✏, Then by Lemma 2.2.6, one can approximate (uniform in x P X an ) the metric FSpk¨k 1 q by continuous metrics FSpk¨k 1 1 q in such a way that distpFSpk¨k 1 q, FSpk¨k 1 1 qq § ✏. If ✏ tends to 0, one can see that the metric FSpk¨k 1 q is also continuous. Lemma 2.2.8.-Assume that L is globally generated. For any metric on L ,a n d any x P X an ,o n eh a s

|p˚qpxq| FSpk¨k q • |p˚qpxq| .
Proof.-For any x P X an and any e P L pxqzt0u, by definition, one has

|e| FSpk¨k q " inf s 1 pxq" e s 1 PR 1 pL q, Pp pxq ˆ| | ´1 x ¨sup yPX an inf s 1 pyq"µt 1 pyq t 1 PR 1 pL q,µPp pxq ˆ|µ| ´1 x kt 1 k • inf s 1 pxq" e s 1 PR 1 pL q, Pp pxq ˆ| | ´1 x ¨inf s 1 pxq"µt 1 pxq t 1 PR 1 pL q,µPp pxq ˆ|µ| ´1 x ¨kt 1 k " |e| so |p˚qpxq| FSpk¨k q • |p˚qpxq| .
Proposition 2.2.9.-Assume that there exist an ultrametric norm k¨k 1 on R 1 pL q such that is given by FSpk¨k 1 q. Then for any n P N, FSpk¨k n q is equal to n .( [ 24, Proposition 3.6])

Proof.-First look at the case n " 1.F o r a n y x P X an and any e P L pxqzt0u,b y construction, one has

|e| FSpk¨k q " inf s 1 pxq" e s 1 PR 1 pL q, Pp pxq ˆ| | ´1 x ¨ks 1 k " inf s 1 pxq" e s 1 PR 1 pL q, Pp pxq ˆ| | ´1 x ¨sup yPX an |s 1 pyq| " inf s 1 pxq" e s 1 PR 1 pL q, Pp pxq ˆ| | ´1 x ¨sup yPX an inf s 1 pyq"µt 1 pyq t 1 PR 1 pL q,µPp pyq ˆ|µ| ´1 y ¨kt 1 k § inf s 1 pxq" e s 1 PR 1 pL q, Pp pxq ˆ| | ´1 x ¨ks 1 k " |e| so |˚| FSpk¨k q pxq § |˚| pxq.
Besides, by Lemma 2.2.8,o n eh a s|˚| FSpk¨k q pxq • |˚| pxq. Hence the two metrics are equal.

Then for general n, notice that by the above equality and Lemma 2.2.8,

|e bn | FSpk¨k n q • |e bn | n " |e| n " |e| n FSpk¨k q ,
But as the family k¨k n ( is sub-multiplicative, the family of quotient norms FSpk¨k n q ( is also sub-multiplicative by Fact 1.3.5, so the reverse inequality holds. Thus the two metrics n and FSpk¨k n q are equal.

Proposition 2.2.10.-Assume that is a semipositive metric on L , then the envelope metric Pp~¨~ q is equal to . (see also [11, Taking limit for n Ñ8and then for ✏ Ñ 0,o n eh a s distpPp~¨~ q, q"0, so the two metrics are equal.

Spectrum and envelope

In this section, let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q, we describe the metric (Berkovich) spectrum of the normed algebra of sections p SpL , ~¨~ q. It is a compact set in the analytification of the algebraic spectrum CpL S' q an . In the first part, we consider the continous map induced by the morphism of k-schemes p an : VpL q an Ñ CpL S' q an , and describe the pre-image of the Berkovich spectrum in VpL q an by the dual unit disc bundle with respect to the envelope metric. In the second part, we roughly locate the position of the spectrum with respect to affinoid domains in CpL S' q an , which contains or is contained in the compact spectrum.

2.3.1.

Dual unit disc bundle. -Recall that the dual unit disc bundle of a metrized line bundle pL , q is the following compact subset of VpL q an D _ pL , q" w P VpL q an , @e 1 P L pxq,

|e 1 | w § |e 1 | ( .
The image p an pD _ pL , qq is a compact set in CpL S' q an , and one denotes it by M ´p q.A point in VpL q an is denoted by w,i t si m a g ei nCpL q an by z and its image in X an by x. -Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q, and let ~¨~be an ultrametric graded algebra norm on S ' pL q. Let w P CpL q an be a point and denote by z P CpL S' q an the point p an L S' . Then z P Mp p SpL , ~¨~qq if and only if one of the following criteria holds 1. there exists Cpzq °0 such that @s P S ' pL q, |s| z § Cpzq¨~s~.

2. there exists Cpzq °0 such that @spxqPS ' pL qpxq, |spxq| w § Cpzq¨~spxq~X |x .

3. there exists C 1 pzq"1 such that @spxqPS ' pL qpxq, |spxq| w § ~spxq~X |x;sp .

where ~¨~X |x;sp is the spectral algebra seminorm of ~¨~X |x .

4. there exists C 1 pzq"1 such that @s P Sym pxq L pxq, |s| w § ~s~1 X|x;sp , where ~¨~1 X|x;sp is the unique graded power multiplicative algebra seminorm on Sym pxq L pxq whose restriction on S ' pL q X|x is ~¨~X |x;sp .

5. there exists C 1 pzq"1 such that @e P L pxq, |e| z § ~e~1 X|x;sp .

there exists

C 1 pzq"1 such that @e 1 P L pxq, |e 1 | z § |e 1 | Pp q .
Or equivalently, w P D _ pL , Pp~¨~qq.

Proof.-The condition 1 unfolds the definition of the Berkovich spectrum. The condition 2 is equivalent to the criterion 1,a s~¨~X |x is the quotient algebra norm of ~¨~p pxq for the evaluation map evpxq. The condition 3 is equivalent to the condition 2 since |¨| z is power-multiplicative.

The conditions 3, 4, 5 and 6 are equivalent: Lemma 2.1.3 implies that the morphism of graded pxq-algebra S' : S ' pL q X|x Ñ Sym pxq L pxq is a (TN)-isomorphim. Proposition 2.1.18 then implies the equivalence of these conditions. Note that Proposition 2.1.23 implies that the algebra seminorm ~¨~1 X|x;sp on Sym pxq L pxq is equal to the seminorm given by envelope pseudometric |¨| Pp~¨~q pxq.

Proposition 2.3.3.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. The restriction of the map p an L S' is a continuous surjective map between topological spaces

D _ pL , Pp~¨~qq Ñ Mp p S ' pL , ~¨~qq A further restriction is an open map D _ pL , Pp~¨~qqzOpXq an Ñ Mp p S ' pL , ~¨~qqz0 0 0 an L S' .
Proof.-By Proposition 2.3.2, the pre-image under p an of Mp p S ' pL , ~¨~qq is exactly D _ pL , Pp~¨~qq, so one has the first restriction map, and it is continuous and surjective.

Propositions 2.1.20 and 2.1.2 indicate that the composition map p an L S' : VpL q an zOpXq an Ñ CpL q an z0 0 0 an Ñ CpL S' q an z0 0 0 an

L S'
is an open map. The second assertion follows easily from general topology.

2.3.3. Spectrum of normed section algebra as geometric envelope. -In previous part we've identified the compact spectrum with the image in the affine cone of the dual unit disc bundle with respect to envelope metric. In this part, we describe it geometrically, as the holomorphic envelope set in the affine cone of the image of the dual unit disc bundle with repect to the original metric. For the abbreviation of notations, we shall henceforth denote by Mp q the compact set p an pD _ pL , Pp qqq and by M ´p q the compact set p an pD _ pL , qq. They are both compact sets contained in CpL S' q an ,a n do n e has M ´p q Ñ Mp q.N o t et h a tMp q is the spectrum of the Banach algebra p R ' pL , q, while M ´p q is not a spectrum of any Banach algebra.

Recall that for any compact set K in an affinoid domain MpAq, denoted by Z, one can construct its holomorphic envelope in Z as the following compact set

p K :" z P Z, @f P A, |f | z § sup z 1 PK |f | z 1 ( .
If A has a dense k-subalgebra A, then one can also use elements in A to define an envelope. It is the same envelope as the one defined by elements of A.

Proposition 2.3.4.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q.W i t h the notations introduced as above, one has

Mp q" { M ´p q,
where the envelope is taken in any affinoid domain in CpL S' q an that contains both.

Proof.-One first shows the inclusion Ö. Since Mp q is the spectrum of p S ' pL , q,i ti s holomorphic convex in CpL S' q an by Corollary 1.4.5. Since it contains M ´p q, it contains also the holomorphic envelope of M ´p q. Conversely, to show the inclusion Ñ, let z P V be a point in { Mp q. By the construction of Mp q and by Lemma 2.3.1, for any s P S ' pL q, one has

|s| z § ~s~ " sup nPN ks n k n ( " sup z 1 PM ´p q |s| z 1 .
As S ' pL q is dense in p S ' pL q, one deduces that z belongs to { M ´p q.

2.3.4. Affinoid domains sandwiching the spectrum. -We show that the Berkovich spectrum Mp p S ' pL q, q is contained in some affinoid domain, and it contains an affinoid domain neighbourhood of 0 0 0 L S' in CpL S' q an . The first affinoid domain will be used to do holomorphic spectral calculus, the second one will be used to deduce the openness of the image in the affine cone of the open dual unit disc bundle. is a homeomorphism from domain to image.

Proof.-This follows directly from Proposition 1.5.5 and Proposition 1.3.7.

Remark 2.3.6.-In particular, if one takes R ' pL X|Y q to be the subalgebra S ' pL q,and ~¨~ ,X|Y to be the algebra norm, then one can construct an affinoid algebra norm ~¨~3 ,X|Y . We shall denote by p RpL X|Y , 3 X|Y q the affinoid Banach algebra p RpL X|Y , ~¨~3 ,X|Y q.

2.3.4.2. From inside. -One can describe the shape of the Berkovich spectrum if the metric is a Fubini-Study metric associated with a norm admitting an orthogonal basis.

Proposition 2.3.7.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. Let n P N be an integer such that L bn is globally generated. Let k¨k n be an ultrametric norm on S n pL q admitting an orthogonal basis ts n,j u jPt0,...,dnu . Let w P VpL q an and z P CpL S' q an be the point p an pwq, then w

P D _ pL , 1 n FSpk¨k n qq (resp.D _ pL , 1 n FSpk¨k n qq)i fa n do n l yi f @j Pt0,...,d n u, |s n,j | z § ks n,j k n presp. † ks n,j k n q.
In particular, the image of D _ pL , 1 n FSpk¨k n qq under p an is an open subset in CpL S' q an . Proof.-By construction of dual unit disc bundle, the point w belongs to D _ pL , So the two conditions are equivalent. Then the openness of the image of D _ pL , 1 n FSpk¨k n qq under p an follows easily from these defining equations.

Corollary 2.3.8.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. Let be a continuous metric on L. Then the image of D _ pL , q under p an is an open subset of CpL S' q an . Proof.-Proposition 2.1.20 implies that the continuous map p an : VpL q an zOpXq an Ñ CpL S' q an z0 0 0 an L S' is an open map. As is continuous, D _ pL , qzOpXq an is an open subset of VpL q an zOpXq an . Then the image of D _ pL , qzOpXq an is an open subset of CpL S' q an z0 0 0 an L S' ,s oi ti sa l s oa n open subset of CpL S' q an .I ts u ffices to find an open neighbourhood of 0 0 0 an L S' in the image of D _ pL , q.N o t et h a t0 0 0 an L S' is the contained in the image of OpXq an . As L is ample, there exist n P N such that L bn is globally generated. Let ts n,j u jPt0,...,dnu be a basis and construct an ultrametric norm k¨k n on S n admitting this basis as an orthogonal basis. Deonte by be the Fubini-Study metric associated with k¨k n .A sb o t h 1 n and are continuous and X an is compact, there exists ✏ P R such that

@x P X an , 1 n p✏qpxq § pxq, so p an pD _ pL , 1 n p✏qqq Ñ p an pD _ pL , qq.
The left hand side is an open subset of CpL S' q an by Lemma 2.3.7. It contains 0 0 0 an L S' . Then the union p an pD _ pL , qq " p an pD _ pL , qzO an pXqq Y p an pD _ pL , 1 n p✏qqq is equal to the image of D _ pL , q. Hence it is an open subset in CpL S' q an . Corollary 2.3.9.-Let S ' pL q be a (TN)-isomorphic subalgebra of R ' pL q. Let be a continuous metric on L .I fPp q is continuous, then for any

✏ P R °0,o n eh a s Mp p S ' pL , qq Ñ Int t C pMp p S ' pL , p✏qqqq,
where Int t C denotes the topological interior as subspace of CpL S' q an .

Proof.-By Proposition 2.3.3, the left hand side Mp p S ' pL , qq is identified with p an pD _ pL , Pp qqq. Since Pp q is continuous, it is contained in the open subset p an pD _ pL , Pp qp✏qqq. This open subset is contained in p an pD _ pL , Pp qp✏qqq which is identified with Mp p S ' pL , p✏qqq, hence p an pD _ pL , Pp qp✏qqq is contained in the topological interior of Mp p S ' pL , p✏qqq, the right hand side.

Restriction to sub-varieties

We study the behavior of norms under the restriction to a sub-variety.

Proposition 2.4.1.-Let be an upper-semicontinuous metric on L . Then Pp~¨~ q| Y " Pp~¨~ ,X|Y q.

CHAPTER 3

NORMED EXTENSION FOR RESTRICTED SECTIONS

In this chapter, given a closed subvariety Y of X and ample line bundle L , we consider the problem of extension of sections of L | Y to X with a control of their norms induced by . We interpret this control as an upper bound of the quotient algebra norm ~¨~X |Y by its spectral algebra norm. We propose two methods in order to establish this inequality.

Background and results

3.1.1. The Problem. -The algebro-geometric setting is as follows: suppose given a sub-variety (reduced closed subscheme) Y of an integral projective scheme X over a field k, and let I Y be the corresponding ideal sheaf. Recall that Serre's vanishing theorem (see for example [56, théorème III.2.2.1]) asserts that there exist an integer n Y such that

H 1 pX, I Y b L bn q"t0u
for any integer n • n Y . Therefore the exact sequence of coherent O X -modules

0 / / I Y b L bn / / L bn / / pO X {I Y qbL bn / / 0
induces a surjective k-linear map from H 0 pX, L bn q to H 0 pY, L | bn Y q, which we denote by ps P H 0 pX, L bn qq fi ›Ñps| Y P H 0 pY, L | bn Y qq, where L | Y denotes the restriction of L to Y . In other words, for any integer n • n Y , any global section of L | bn Y extends to a global section of L bn .O n es a y st h a ts n is an extension of t n . Now consider the metric analogue of the above statement. Assume that k is equipped with a complete absolute value |¨| and let L be equipped with a continuous metric . Denote by L | Y and | Y their restrictions to Y , respectively. The metric extension problem is to find an extension s n P H 0 pX, L bn q of a given prescribed section t n P H 0 pY, L | bn Y q of the restricted line bundle, such that the distorsion of their supremum norms is smallest possible. In terms of norms, the problem can be interpreted as looking for a uniform upper bound for [START_REF] Bedford | The Dirichlet problem for a complex Monge-Ampère equation[END_REF] inf

sPH 0 pX,L bn q s| Y "t ksk n ktk n | Y ,t P ImpH 0 pX, L bn q | Y ݄ H 0 pY, L | bn Y qqzt0u.
which can be further formulated as the comparison of two norms on one single vector space

ImpH 0 pX, L bn q | Y ݄ H 0 pY, L | bn
Y qq, one is the quotient norm of k¨k n , denoted by k¨k n ,X|Y ; another is the sup norm k¨k n | Y .

In practice, one contents with knowing just the asymptotic behavior (when n Ñ8)o f such upper bounds.

3.1.2. Complex case. -In the complex analytic setting, namely when pk, |¨|q is C equipped with the usual absolute value, the metric extension problem has been studied by different authors using various approaches. Assume that the metric is strictly positive (namely, for any local section s of L over an open subscheme U of X, the function px P U an q fi Ñ log|spxq| is strongly plurisubharmonic). In the case where X is smooth, by Gromov's theorem we can compare the sup norm k¨k n to the L 2 norm k¨k n ,L 2 defined as

@ s P H 0 pX, L bn q, ksk n ,L 2 :" ˆªX an |spxq| n pxq dV ˙1 2 ,
where dV is a probability measure on X an which is locally equivalent with Lebesgue measure with a smooth Radon-Nikodym density. Therefore, in the case where X and Y are both smooth, we can apply the Hörmander L 2 technique and get an inequality (see for example [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF], [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF], [START_REF] Manivel | Un théorème de prolongement L 2 de sections holomorphes d'un fibré hermitien[END_REF], [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF])

(3) k¨k n | Y • C 1 p ,Y,Xqn ´dk¨k n ,X|Y ,n • n Y ,
where C 1 p ,Y,Xq is a positive constant. Alternatively, one can apply Grauert's argument of pseudo-convexity of the (open) dual unit disc bundle of pL , q to produce, for any ✏ °0, a slightly weaker inequality of the form which however does not require the smoothness of the schemes X and Y (4)

k¨k n | Y • C ✏ p ,Y,Xqe ´✏n k¨k n ,X|Y ,n • n Y ,
where C ✏ p ,Y,Xq is a positive constant depending on ✏. We refer to [START_REF] Bost | Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems[END_REF]and [START_REF] Randriambololona | Métriques de sous-quotient et théorème de Hilbert-Samuel arithmétique pour les faisceaux cohérents[END_REF] for more details.

Note that, to obtain an estimate of the form (4) for any closed point Y , the semipositivity of the metric (in the sense of uniform approximability by Fubini-Study metrics) is actually necessary. In fact, for sufficiently positive integer n, the ample invertible O Xmodule determines a closed embedding ◆ n : X Ñ PpH 0 pX, L bn qq. The norm k¨k n on H 0 pX, L bn q defines a Fubini-Study metric on the universal invertible sheaf of the projective space H 0 pX, L bn q. Denote by n the continuous metric on L the n-th tensor power of which identifies with the restriction of the Fubini-Study metric. Then the estimate (4) in the case where Y is a single closed point txu implies that the sequence of norms |¨| n pxq converges to |¨| pxq. Moreover, it can be shown the subsequence p|¨| 2 m pxqq mPN is decreasing. Therefore, the semipositivity of Fubini-Study metrics implies that of the metric .

3.1.3. Non-Archimedean case. -The problem of normed extension has various applications, not only in complex analytic geometry, but also in Arakelov geometry. It is a key ingredient in the proof of the arithmetic Hilbert-Samuel theorem, see [START_REF] Abbes | Théorème de Hilbert-Samuel "arithmétique[END_REF]. It has also been applied in the proof of Nakai-Moishezon criterion of arithmetic ampleness, cf. [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF], see also [START_REF] Gubler | Local heights of subvarieties over non-Archimedean fields[END_REF].

From the adelic point of view of Arakelov geometry, one can replace the integral models of arithmetic objects by a family of Berkovich analytic objects (possibly equipped with metrics) parametrised by the set of finite places of a number field. The advantage of the adelic approach consists in treating all places of a number field in a uniform way. This motivates the research of the non-Archimedean analogue of notions and results in complex analytic geometry. It turns out that many usual analytic tools (such as the L 2 method for B) do not work well in the non-Archimedean setting (see however [START_REF] Stevensson | A NON-ARCHIMEDEAN OHSAWA-TAKEGOSHI EXTEN-SION THEOREM[END_REF] for a non-Archimedean Ohsawa-Takegoshi extension theorem in one dimension), and often new ideas are needed to develop the non-Archimedean analogue of complex analytic geometry and to unify the arguments in both settings.

In this chapter we undertake a study of the metric extension problem in (2)inthenon-Archimedean analytic setting. Previously, a metric extension property of the following form is established in [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF], Theorem 3.1.1 (Theorem 3.2.1). -Let be a semipositive metric on L , then for any ✏ °0,a n da n yt 1 P H 0 pY, L | Y q, there exists n Y P N such that for any n • n Y , there exist s n P H 0 pX, L bn q with s n | Y " t bn 1 and

ks n k n § e n✏ ¨pkt 1 k | Y q n .
A metric which is a uniform limit of model metrics is called "semi-amply metrized" in [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF]. For any model metric associated with a model pX, Lq of pX, L q for o k Ñ k,i n [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF] they look at the normed vector spaces of global sections H 0 pX, L q with the norm given by a lattice H 0 pX, Lq in it. By a careful study of fine properties of norms in a vector space associated with lattices, they first established this metric extension for such model metrics. Then by the existence of non-Archimedean (almost)-orthogonal basis for arbitrary ultrametric norm, they can approximate a general Fubini-Study metric by model metrics, thus getting the final conclusion. As a by-product, they showed that one may use general Fubini-Study metrics (coming from general ultrametric norms on H 0 pX, L q) instead of model metrics in the approximation process. Thus on an ample line bundle, a metric that is"semi-amplely metrized" in [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF] is the same as being asymptotically Fubini-Study in [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF].

Thanks to this equivalence, we can avoid the use and study of models. Our new ingredient to this metric extension problem is to investigate the normed algebra of sections instead of normed vector spaces of sections. From point of view of normed section algebras, the goal is to compare two algebra norms ~¨~ ,X|Y and ~¨~ Y on the graded algebra

R ' pL X|Y q :" à nPN Im `H0 pX, L bn q | Y ›Ñ H 0 pY, L | bn Y q ˘.
We would like to bound from the above of the quotient one by the sup one. The advantage of passing from vector spaces to an algebra is to better exploit the algebra structure and to work in an affine geometric setting than in a projective geometric one.

By virtue of Corollary 2.4.4, we know that the sup algebra norm is the spectral norm of the quotient algebra norm. This point of view is the leading thread of our methods: it suffices to concentrate on the relation between the quotient algebra norm ~¨~ ,X|Y and its spectral norm. First, in §1, Corollary 2.4.4 allows us to derive directly the normed extension theorem in [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF]: fix an error of precision ✏, one shows that there is a subexponential bound e n✏ for n • N with N depending on the geometry (X, Y , L and ) and also possibly on the choice of restricted section.

We can also refine the result, dropping the dependence of N on the choice of restricted section so that the bound becomes uniform in this choice, which is our main result of this chapter Theorem 3.1.2 (Theorem 3.3.5,Theorem3.4.10). -Let be a semipositive metric on L . Then for any ✏ °0, there exists n Y P N such that for any n • n Y and any t n P H 0 pY, L | bn Y q, one can find s n P H 0 pX, L bn q such that s n | Y " t n and

ks n k n § e n✏ ¨kt n k n | Y .
The refinement requires a finer comparison result: any (reduced) affinoid algebra norm is equivalent to its spectral norm. We devise two methods of approximation of ~¨~ ,X|Y by affinoid algebra norms, so as to refine the normed extension results in §2 and §3.

Note that our methods give a sub-exponential upper bound as in [START_REF] Berman | Growth of balls of holomorphic sections and energy at equilibrium[END_REF]. Whether the polynomial bound of ( 3) is obtainable in the non-Archimedean setting remains unexplored, and evidently it requires considerations of intrinsic geometry in terms of metrized (co-)tangent sheaves and local parameters. It seems that the commutative Banach algebra techniques that we used here are insufficient to ameliorate the bound as only extrinsic geometry (i.e. embeddings into projective space) has been probed.

Comparison of norms

Recall that we shall compare two algebra norms ~¨~ ,X|Y and ~¨~ | Y on the restricted section algebra

R ' pL X|Y q :" à nPN ImpH 0 pX, L bn q | Y ›Ñ H 0 pY, L | bn Y qq

Geometric approximation

Mp p R ' pL X|Y , X|Y qq Ñ W ✏ Ñ Mp p R ' pL X|Y , | Y p✏qqq Ñ Mp p R ' pL X|Y , 3 ✏ q 3.3.
A f / / i W ✏ ✏ B W ✓ > > .
One can apply Proposition 3.3.2 for ✏ 1 to construct a special domain

W ✏ 1 such that Mp p R ' pL X|Y , X|Y qq Ñ W ✏ 1 Ñ Mp p R ' pL X|Y , | Y p✏ 1 qqq. Proposition 3.3.4.
-For any ✏ 1 °0, there exist a homomorphism of Banach k-algebras

✓p✏ 1 q : W ✏ 1 Ñ p R ' pL X|Y , X|Y q
which is induced by the identity map on the dense sub-k-algebra R ' pL X|Y q.

Proof.-By Lemma 3.3.1, for any ✏ 1 °0, one can construct an affinoid algebra norm ~¨~3 ✏ 1 p✏ 1 q,X|Y on R ' pL X|Y q such that the corresponding affinoid algebra admits a (continuous) homomorphism to the original Banach algebra p R ' pL X|Y , 3 ✏ 1 qÑ p R ' pL X|Y , X|Y p✏ 1 qq.

Composing it with the continuous homomorphism p R ' pL X|Y , X|Y p✏ 1 qq Ñ p R ' pL X|Y , X|Y q, one gets a (continuous) homomorphism from an affinoid algebra to a Banach algebra

⌧ p✏ 1 q : p R ' pL X|Y , 3 ✏ 1 qÑ p R ' pL X|Y , X|Y q.
Its restriction on the dense subalgebra R ' pL X|Y q is the identity, so this homomorphism has dense image, thus by Proposition 1.2.11 the induced continuous map

⌧ p✏ 1 q ˚: Mp p R ' pL X|Y , X|Y qq Ñ Mp p R ' pL X|Y , 3 ✏ 1 qq
is injective and is closed. As both spaces are compact and Hausdorff,t h i sm a pi sa homeomorphism of topological spaces from its domain to its image. So the spectrum Σ ⌧ p✏ 1 q (of the homomorphism ⌧ p✏ 1 q) can be identified with Mp p

R ' pL X|Y , X|Y qq, it is contained in W ✏ 1 .
One can perform spectral calculus for the homomorphism ⌧ p✏ 1 q with respect to the special domain W ✏ 1 ,a sΣ ⌧ p✏ 1 q is contained in W ✏ 1 . By Theorem 1.4.8, there exists a homomorphism ✓p✏{2q of Banach k-algebras W ✏ 1 Ñ p R ' pL X|Y , X|Y q which is induced by the identity map on the dense sub-k-algebra R ' pL X|Y q, and makes the following diagram commutative p R ' pL X|Y , 3p✏ 1 qq

⌧ p✏ 1 q / / i W ✏ 1 ✏ ✏ p R ' pL X|Y , X|Y q W ✏ 1 ✓p✏ 1 q 5 5
. Theorem 3.3.5.-Let be a semipositive metric on L . Then for any ✏ °0, there exist n Y P N such that for any n • n Y and any t n P H 0 pY, L | bn Y q, one can find s n P H 0 pX, L bn q such that s n | Y " t n and

ks n k n § e n✏ ¨kt n k n | Y .
Proof.-First, fix some ✏ 1 † ✏ and one constructs the homomorphism ✓p✏ 1 q W ✏ 1 as in Proposition 3.3.4. The boundedness (continuity) of this homomorphism of Banach kalgebras implies that there exist C ✏ 1 °0 such that

@t P R ' pL X|Y q, ~t~ ,X|Y § C ✏ 1 ¨~t~W ✏ 1 " C ✏ 1 ¨sup zPW ✏ 1 ¨|t| z .
Second, by Proposition 2.4.3, one has a canonical homeomorphism

Mp p R ' pL X|Y , p✏ 1 q X|Y qq » Mp p R ' pL X|Y , | Y p✏ 1 qqq
from which one deduces

W ✏ 1 Ñ Mp p R ' pL X|Y , | Y p✏ 1 qqq.
Remember that since ~¨~ | Y p✏ 1 q is power-mutliplicative, by Theorem 1.2.14, it is the supremum norm on Mp p R ' pL X|Y , | Y p✏ 1 qqq. Hence by comparing supremum norms on these two closed sets, we get

@t P R ' pL X|Y q, ~t~W ✏ 1 § ~t~ | Y p✏ 1 q .
Combining these two inequalities, for any t n P R n pL X|Y q,o n eh a s

kt n k n ,X|Y "~t n ~ ,X|Y § C ✏ 1 ¨~t n ~W✏ 1 § C ✏ 1 ¨~t n ~ | Y p✏ 1 q " C ✏ 1 ¨en✏ 1 ¨kt n k n | Y .
Let n Y be the integer maxtrlogpC ✏ 1 q{p✏ ´✏1 qs,Mu, then for any n • n Y and any t n P H 0 pY, L | bn Y q, by the definition of quotient norm, one can find s n P H 0 pX, L bn q such that

ks n k n § e n✏ ¨kt n k n | Y . Remark 3.3.6.
-The continuity of is necessary in this approach: it is needed in Proposition 3.3.2 to guarantee that there are enough space between Mp p R ' pL X|Y , X|Y qq and Mp p R ' pL X|Y , | Y p✏qqq so as to insert a special domain W ✏ . Otherwise, the two domains may touch at some boundary point and there would be no special domain in the larger one that is an e i g h b o u r h o o do f the smaller one.

Algebraic approximation

In this part, we approximate the algebra norm ~¨~ X|Y by one affinoid algebra norm on R ' pL X|Y q globally. This algebraic approximation exploits subtle consequences of existence of a ultra-metric orthogonal basis in pR n pL q, k¨k n q for some n P N. In this section pk, |¨|q is assumed to be discretely valued.

Recall that one denotes by Hpk, |¨|q the Q-vector space generated by log| | ( Pk ˆ,a n d one says that a multi-radius r r r "p r 0 ,...,r d q is pk, |¨|q-free if the images of the numbers tlog r i u iPt0,...,du in the quotient Q-vector space R{Hpk, |¨|q is Q-linearly independent. Recall that thanks to the discreteness of |¨|, any finite dimensional ultrametric normed k-vector space has an orthogonal basis. (see Proposition 1.1.16) Our goal is to deduce an upper bound of ~¨~ ,X|Y by (some variant of)~¨~ Y . Recall that the second algebra norm is the spectral norm of the first, thanks to Corollary 2.4.4.

The key is to note that such an upper bound of an algebra norm by its spectral norm holds whenever the algebra norm is an affinoid algebra norm. Namely, if ~¨~ ,X|Y is an affinoid algebra norm, by Corollary 1.3.22 there exists C P R °0 such that

~¨~ X|Y § C ¨~¨~ X|Y ;sp " C ¨~¨~ Y .
An explicite calculation of ~¨~ is possible in the case where X is P d , L is Op1q and is a (genuine) Fubini-Study metric (induced from an ultrametric norm on R 1 pOp1qqq.I n this case ~¨~ is a Gauss-Tate norm of some multiradius. This implies that for general X, if L is very ample and is a (first level) Fubini-Study metric, the algebra norm ~¨~ ,X|Y is an affinoid algebra norm, thus the desired upper bound can be obtained in this case. It would be nice if this affinoid property also holds whenever L is just ample and is some m-th Fubini-Study metric, because one can treat the general case of an semipositive metric using approximations. Unfortunately we are unable to show the affinoid property of ~¨~ ,X|Y in this m-th Fubini-Study metric case. -Let be a metric on Op1q, one studies the algebra norm ~¨~ on R ' pOp1qq. Since R ' pOp1qq can be identified with the multivariate polynomial algebra, one would like to show that under various assumptions, the algebra norm ~¨~ can be identified with a Gauss-Tate algebra norm on this multi-variate polynomial algebra. As a consequence, the normed section algebra pR ' pOp1qq, ~¨~ q will be a Tate affinoid algebra. (see Definition 1.3.1)

First, one chooses a preferred basis to do the identification. By Proposition 1.1.16,o n e can find an orthogonal basis tT i u iPt0,...,du for the normed vector space pR 1 pOp1qq, k¨k q, namely a basis of the k-vector space R 1 pOp1qq satisfying @p 0 ,..., d qPk d`1 ,

d ÿ i"0 i ¨Ti " max iPt0,...,du | i | ¨kT i k ( .
For any i Pt0,...,du, one denotes by r i the value kT i k ,a n db yr r r PpR °0q d`1 the multiradius pr 0 ,...,r d q. One fixes such an orthogonal basis, and identify the graded k-algebras

R ' pOp1qq " Sym R 1 pOp1qq » krT 0 ,...,T d s,T i fi Ñ T i .
For any multi-index J "p j 0 ,...,j d qPN d`1 , one denotes by T T T J the monomial element ± iPt0,...,du pT i q j i P R |J| pOp1qq,a n db yR J pOp1qq is the one dimensional subspace generated by the monomial section T T T J . With this choice of basis, one has a multi-grading by J P N d`1 on the section algebra which refines the original grading by n P N

R ' pOp1qq " à nPN R n pOp1qq » à nPN à JÑN d`1 ,|J|"n R J pOp1qq " krT 0 ,...,T d s.
Second, one would like to identify the algebra norm ~¨~ . Note that in general, since ~¨~ is graded (with respect to the grading n P N), the sub-spaces R n pOp1qq are orthogonal with respect to ~¨~ for different n P N. However, a Gauss-Tate algebra norm exhibits an orthogonality with respect to the finer multi-grading J Ñ N d`1 . In fact, for an algebra norm to be Gauss-Tate, the subspaces

R J pOp1qq ( JPN d`1
should also be orthogonal. We shall show that ~¨~ coincide with a Gauss-Tate algebra norm ~¨~T r r rp0 0 0q of multiradius r r r on R ' pOp1qq, in two steps:

1. The two norms coincide on each subspace R J pOp1qq;

2. The two norms coincide on each subspace R n pOp1qq, hence also on R ' pOp1qq.

First, one calculates ~¨~ on mononial elements.

kT i k j i .
Proof.-Take a complete non-Archimedean valued field extension pk 1 , |¨| 1 q of pk, |¨| k q such that @i Pt0,...,du,r i P |k 1ˆ|1 , hence for any i Pt0,...,du, there exist elements  i P k 1 such that | i | 1 " r i . One denotes by xpr r rqPpP d k q an the point given by the coordinates r 0 : ... :  d s. Claim 3.4.2.-For any i Pt0,...,du,o n eh a s

kT i k " r i " |T i | pxpr r rqq.
In other words, the maximum of the function |T i | pxq on pP d k q an is r i ,a n dt h em a x i m u m values of these d `1 functions on pP d k q an can be attained at the same point xpr r rq. Proof.-With the orthogonality of the basis tT i u iPt0,...,du , we can calculate as follows

|T i | pxpr
! | m || m | ) " inf 0 p 0 { i q`¨¨¨` d p d { i q"1 p 0 ,..., d qPk d`1 max mPt0,...,du ! | m || m { i | ¨| i | ) " | i | " r i .
The last equality is obtained by Lemma 2.2.2.

By this Claim, for any multi-index J, the function |T T T J | pxq can attain its maximum value ± iPt0,...,du r j i i at the point xpr r rqPpP d k q an as the product of maxima of the one-variate factors T j i i ( iPt1,...,du . By definition of sup norm, one has kT T T J k |J| " sup

xPpP d q an |T T T J | pxq" π iPt0,...,du r j i i " π iPt0,...,du kT i k j i .
Second, one calculates the algebra norm ~¨~ on any (homogeneous) combination of monomials. For a general metric, one needs a pk, |¨|q-freeness assumption on the multiradius r r r to gain finer orthogonality.

P k,o n eh a s ⌫ ⌫ ÿ JPS J ¨T T T J ⌫ ⌫ " sup JPS k J ¨T T T J k |J| .
In other words, the algebra norm ~¨~ on R ' pOp1qq can be identified with the Gauss norm on krT 0 ,...,T d s of multi-radius r r r. The Banach k-algebra p R ' pOp1q, q is an affinoid algebra.

Proof.-By the pk, |¨|q-freeness assumption on r r r and by Proposition 3.4.1, for any two distinct multi-index J and J 1 , and any two non-zero coefficients J and J 1 in k,w eh a v e

J ¨T T T J |J| " | J | ¨π iPt0,...,du r j i i ‰ | J 1 | ¨π iPt0,...,du r j 1 i i " J 1 ¨T T T J 1 |J 1 | .
By Lemma 1.1.13, the elements tT T T J u JPS form an orthogonal basis for the normed vector space p À JPS k ¨T T T J , ~¨~ q. So the desired equality holds. Corollary 3.4.4.-With the same hypothesis, the envelope metric Pp q is a Fubini-Study metric induced by k¨k , and it is continuous.

If is a Fubini-Study metric, one does not need the pk, |¨|q-freeness assumption on r r r. For any p 0 ,..., d qPR d`1 , denoted by , one can construct a perturbed metric p q as follows. Let k¨k p q be the following ultrametric norm on R 1 pOp1qq @p 0 ,...,

d qPk d`1 , d ÿ i"0 i ¨Ti " max iPt0,...,du e i | i | ¨kT i k ( ,
namely the ultrametric norm such that tT i u iPt1,...,du is an orthogonal basis, and that @i Pt0,...,du, kT i k p q " e i kT i k .

Let p q be the metric FSpk¨k p q q on Op1q.D e n o t eb y| | the number max iPt0,...,du | i | P R °0.

Lemma 3.4.5.-Assume that is a Fubini-Study metric. For any ✏ °0, there exist P R d`1 with | | § ✏ such that @n P N, distpk¨k n , k¨k n p q q § n✏.

Proof.-Choose an arbitrary with | | § ✏. Then distpk¨k , k¨k p q q § ✏.

By Proposition 2.2.10,w eh a v e distpFSpk¨k q, FSpk¨k p q qq " distpFSpk¨k q, p qq § ✏, by the assumption and Proposition 2.2.9, FSpk¨k q" , so the conlusion holds.

J ¨T T T J n p q § ÿ |J|"n J ¨T T T J n § e n✏ ÿ |J|"n J ¨T T T J n p q . By Proposition 3.4.3,o n eh a s max |J|"n e ´n✏ | J | π iPt0,...,du pe i r i q j i ( § ÿ |J|"n J ¨T T T J n § max |J|"n e n✏ | J | π iPt0,...,du pe i r i q j i ( .
Fix n and let ✏ tends to 0, one gets ÿ

|J|"n J ¨T T T J n " max |J|"n | J | π iPt0,...,du r j i i ( .

Finally one deduces that

⌫ ⌫ ÿ |J| †8 J ¨T T T J ⌫ ⌫ n " sup nPN max |J|"n | J | π iPt0,...,du r j i i ( " max |J| †8 | J | π iPt0,...,du r j i i ( .
Hence ~¨~ can be identified with a Gauss-Tate norm of multi-radius r r r on R ' pOp1qq. Proof.-By the very-ampleness assumption, the space of global sections R 1 pL q induces an projective embedding

◆ 1 : X Ñ P d 1
´1 k such that ◆ 1 Op1q is L . Moreover there exist a norm k¨k 1 on R 1 pL q such that is given by FSpk¨k 1 q. One can view k¨k 1 as a norm on R 1 pOp1qq, and one gets a metric " FSpk¨k 1 q on Op1q. By construction the restriction metric | X coincides with .

By Proposition 3.4.6, the Banach algebra p R ' pOp1q, ~¨~ q is an affinoid algebra. The embedding ◆ 1 induces a quotient homomorphism of Banach algebras

p R ' pOp1q, ~¨~ qÑ p R ' pL , ~¨~ ,P d 1 |X q.
Hence the quotient Banach algebra p R ' pL , ~¨~ ,P d 1 |X q is an affinoid algebra. By Proposition 2.4.4,on p R ' pL , ~¨~ ,P d 1 |X q, the algebra norm ~¨~ is the spectral norm of ~¨~ ,P d 1 |X . The later algebra norm is an affinoid norm, since this Banach algebra is reduced, this norm is equivalent to its spectral norm by Proposition 1.3.22.S o~¨~ is also an affinoid algebra norm. Thus p R ' pL , q is an affinoid algebra. By the same argument, one has the quotient homomorphism of Banach algebras

p R ' pL , qÑ p R ' pL X|Y , X|Y q,
As ~¨~ is affinoid algebra norms, so is ~¨~ ,X|Y . Then on p R ' pL X|Y , ~¨~ ,X|Y q,b yProposition 2.4.4, the algebra norm ~¨~ | Y is the spectral norm of ~¨~ ,X|Y , hence is itself an affinoid algebra norm. Proposition 3.4.8.-Assume that is a (first level) Fubini-Study metric and that L is very ample. Then there exists Cp ,Yq °0 such that for any t P R ' pL X|Y q, one can find s

P R ' pL q satisfying ~s~ § Cp ,Y,Xq¨~t~ | Y .
In particular, for any n P N and t n P R ' pL X|Y q, one can find s n P R ' pL q satisfying

ks n k n § Cp ,Y,Xq¨kt n k n | Y .
Proof.-By Proposition 3.4.7 and the assumption, the Banach algebra norm ~¨~ X|Y is an affinoid algebra norm. As p R ' pL X|Y , X|Y q is reduced by Proposition Proof.-By the semipositivity assumption, there exist norms tk¨k n u nPN on R n pL q such that is given by Pptk¨k n u nPN q. Let n denote the metric 1 n FSpk¨kq n on L . Since is continuous, the convergence of t n u nPN to the envelope metric is uniform. So for any ✏ °0, there exist

M 1 • M such that distp M 1 , q § 1 3 ✏,
hence for any n P N,o n eh a s

distpk¨k n M 1 , k¨k n q § 1 3 n✏, distpk¨k n M 1 ,X|Y , k¨k n ,X|Y q § 1 3 n✏.
By our assumption, the Banach algebra p R ' pL X|Y , p M 1 q X|Y q is an affinoid algebra, thus there exist C M 1 pX, Y, q °0 such that @t P R ' pL X|Y q, ~t~ M 1 ,X|Y § C M 1 ¨~t~ M 1 ,X|Y ;sp "~t~ M 1 ,Y . Combining this comparison with above estimates, for every n • n Y :" rlnpC M 1 q{p✏{3qs, one has

kt n k n ,X|Y § e 1 3 n✏ ¨kt n k n M 1 ,X|Y § e 1 3 n✏ ¨CM 1 ¨kt n k n M 1 | Y § e 2 3 n✏ ¨CM 1 ¨kt n k n | Y § e n✏ ¨kt n k n | Y .
Hence there exist s n P R n pL q with ks n k n § e n✏ ¨kt n k n | Y . Remark 3.4.11.-The continuity assumption on is necessary in this approach, for deducing a uniform convergence to by m-th Fubini-Study metrics.

CHAPTER 4 CONTINUITY OF ENVELOPE METRIC

In this chapter we study the continuity of the semipositive envelope metric Pp q of a continuous metric on the ample line bundle L . We interpret the continuity of Pp q as a neighbourhood property for the holomorphic convex set Mp q. This helps to reduce the continuity problem to a non-Archimedean version of Cartan-Thullen's result about holomorphic convexity.

Background and results

4.1.1. The Problem. -We maintain the basic setting as in previous chapters. Let k be a field, X be an integral projective scheme over Spec k, L be an ample line bundle over X. The algebraic positivity of L plays a central role in establishing vanishing theorems and their various consequences. If k is equipped with a complete absolute value |¨|, one can construct the corresponding analytic space and put a continuous metric "p|¨| pxqq xPX an on the analytification L an . It turns out that the algebraic positivity of L can be captured by the metric positivity of the pair pL , q, and many problems concerning (algebraic) positive line bundle can be studied via analytic methods with the help of a positive metric .

In case where the metric is not positive, there is a way to construct a positive metric with some positivity from itself. One builds a sequence of positive metrics n on L bn from , and takes the limit metric of the (normalized) sequence of these positive metrics. The limit is a metric on L , and it still remains to be positive (in some sense). In both Archimedean and non-Archimedean settings, Fubini-Study metrics are often used as models for n : they are constructed from an (in fact any) orthogonal basis ts n,j u jPt0,...,dnu of the normed vector space pH 0 pX, L bn q, k¨k n q.I ft h en o r m sk¨k n are chosen to be the norms induced by n (the L 2 -norm with respect to a fixed volume form on X an in the Archimedean setting and the supremum norm in the non-Archimedean setting), then the limit metric is called the Fubini-Study envelope metric of , and is denoted by Pp q.A st h e positivity of Pp q is inherited from the positivity of global objects (i.e. the Fubini-Study metrics), one shall call it a global positivity.

While inheriting the global positivity from Fubini-Study metrics, the envelope metric Pp q may not be continuous on X an . In general, the approximation procedure only yields a possibly singular metric, as the convergence to the limit metric is just pointwise on each fiber of L an . The continuity of envelope metric is important for many problems. For example, when X is proper, the continuity of envelope metric is equivalent to the uniformity of the approximation of Pp q by Fubini-Study metrics n . This uniformity of approximation is indispensable to establishing uniform metric extensions for restricted sections of L from a sub-variety Y of X (see the previous chapter). Another example concerns some functionals which are defined on the space of all continuous metrics such as the relative metric volume. They are devised in search for a canonical metric on L , and there is a need to put Pp q as an argument in these functionals.

4.1.2. Complex case. -In the complex analytic setting, the positivity of a metric is defined by local objects, as the pluri-subharmonicity (psh) of the norm function |epxq| of any local regular section e of L . This local positivity of implies the global positivity of [START_REF] Tian | On a set of polarized Kähler metrics on algebraic manifolds[END_REF][62, Theorem 3.5], namely that is a pointwise limit of Fubini-Study metrics. Concerning the regularity, a deep result of regularization of psh functions using L 2 methods for B from [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF] implies that Theorem 4.1.1.-Suppose that X is normal. If is continuous, then Pp q is also continuous. ( [START_REF] Demailly | Regularization of closed positive currents and intersection theory[END_REF], see also [START_REF] Guedj | Intrinsic capacities on compact Kähler manifolds[END_REF]Appendix]) 4.1.3. Non-Archimedean case. -4.1.3.1. Previous results: "Zariski" approach. -In the non-Archimedean case, semipositivity of a metric is first defined in [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF] using model metrics in the approximation. For various other definitions of semi-positivity, we refer to [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF] for a clear discussion. In this vein, the envelope metric continuity has been studied in [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF], [START_REF] Gubler | Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals (with an Appendix by José Ignacio Burgos Gil and Martín Sombra)[END_REF]a n d [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF]. In particular, the following cases are known from these works. Theorem 4.1.2.-Suppose that X is normal. Suppose that the base field k is of equicharacteristic 0, and the valuation is discrete or trivial. If is continuous, then Pp q is also continuous. ( [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF], [START_REF] Boucksom | Singular semipositive metrics on line bundles on varieties over trivially valued fields[END_REF]) Theorem 4.1.3.-Suppose that X is normal. Suppose that the base field k is of positive equi-characteristic p,andthev aluation|¨| is discrete. Under the hypothesis of the existence of a resolution of singularity in characteristic p,i f is continuous, then Pp q is also continuous. ( [START_REF] Gubler | Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals (with an Appendix by José Ignacio Burgos Gil and Martín Sombra)[END_REF]) These works adopt the "Zariski" point of view to the study of analytic spaces, emphasizing the role of integral models and using algebro-geometric techniques. In these works, the semipositivity of a metric is defined by approximation by model metrics, in the sense of [START_REF] Bloch | Non-Archimedean Arakelov theory[END_REF]and [START_REF] Zhang | Positive line bundles on arithmetic varieties[END_REF]. In [START_REF] Boucksom | Singular semipositive metrics in non-Archimedean geometry[END_REF], a careful analysis of behavior of different models is carried out, and a wonderful framework of non-Archimedean pluripotential theory was develpped. Their general strategy is to analyze the behavior of metrics on the subset of X an consisting of quasi-monomial points. First, a sequence of vertical ideals is associated with an SNC model for pX, L q, and a sequence of model metrics is thus constructed. Then the uniform convergence of this sequence of metrics to the envelope metric is captured by a sequence of multiplier ideals constructed from the integral models. Further properties of multiplier ideals gives control on the approximation and finally lead to the uniform convergence. Additional hypothesis on the base field are imposed in order to facilitate the use of multiplier ideals and of SNC models.

In [START_REF] Gubler | Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals (with an Appendix by José Ignacio Burgos Gil and Martín Sombra)[END_REF], some of the assumptions on the base field has been removed. They can also treat the equal positive characteristic case. Hypothesis of resolution of singularities is imposed to deal with SNC models, and multiplier ideals are replaced by test ideals for the control of approximation.

In view of these results, it is conjectured in [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF]Conjecture 6.20] that Pp q can inherit the continuity of , in a general setting without much further assumptions on the valued base field. Other implications related to this conjecture concerning regularization of non-Archimedean psh functions can be found in [START_REF] Boucksom | Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry[END_REF]Section 6].

4.1.3.2. Our proposal: "Berkovich" approach. -In this chapter we would like to study the continuity problem from the "Berkovich" point of view, emphasizing the role of analysis of holomorphic functions. We propose an analytic route, which is independent of the aforementioned algebro-geometric methods. We reduce this problem to a property of holomorphic convex sets in non-Archimedean analytic space. Namely, we believe that the following assertion should hold: Speculation 4.1.4 (4.2.1). -Suppose that pk, |¨|q is spherically complete. Let Z be the analytification of a normal affine variety. For any compact set K and U in Z with K î U ,o n eh a s p K î p U . In other words, for and any neighbourhood U of K in Z,t h e envelope p U is a neighbourhood of the envelope p K. (The holomorphic envelope is taken in Z.)

The main result establishes the implication of the above property to the continuity problem.

Theorem 4.1.5.-Suppose that the base valued field pk, |¨|q is spherically complete. Suppose that X is normal. Assume that Speculation 4.2.1 is true. Then if is continuous, the envelope metric Pp q is also continuous. Remark 4.1.6.-Valued fields as Q p or F p pptqq, verify the assumptions. On the contrary, C p is a complete valued field that is not spherically complete.

We study the continuity of the envelope metric by the geometry of holomorphic convexity. Our approach is quite different from previous ones. We benifit from [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF]t ou s ea n equivalent definition of semi-positivity via approximation by general Fubini-Study metrics (instead of those which are model metrics). Thanks to this relaxation, our analysis does not concern with integral models. Further, we investigate the envelope metric from the point of view of normed algebra of global sections (instead of the sequence of normed vector spaces of sections). The basic construction is as follows, which relates the envelope metric with the spectrum of normed section algebra (see chapter II for the details).

-Normed algebra of sections. For an ample line bundle L over a projective variety X, we consider the graded algebra of sections R ' pL q" à nPN R n pL q :" à nPN H 0 pX, L bn q,

If L an is equipped with an upper-semicontinuous metric, we construct a graded sub-multiplicative norm on this algebra ~ps n q~ " sup nPN ks n k n where k¨k n are sup norms on H 0 pX, L bn q with respect to the metric n . We can consider a Banach algebra, the completion of the algebra of sections with respect to this norm, which is named as the normed algebra of sections. This Banach algebra is denoted by p R ' pL , q.

-Spectrum. The Zariski spectrum of R ' pL q is the affine cone of L and is denoted by CpL q. It can be identified with the total space of L _ , denoted by VpL q, with the zero section contracted to the origin by a morphism p.I n s i d eCpL q an one finds a compact set, the Berkovich spectrum of p R ' pL , q, denoted by MpL , q. The corresponding compact set of pre-image of MpL , q in VpL q an is exactly the dual unit disc bundle of L with respect to the envelope metric, denoted by D _ pL , Pp qq.

One may also look at the smaller dual unit disc bundle D _ pL , q with respect to the original metric, and denote by M ´pL , q its image under p an . The spaces and relations are described by the following Cartesian digram for maps between topological spaces, where p an is the analytic morphism of contraction of the zero section. From this result, one can recover the envelope metric Pp q directly from MpL , q, which is itself constructed from in a simple way. It then suffices to study the envelope metric from the geometry of these associated sets. Indeed, the global positivity of the metric Pp q is translated to the holomorphic convexity of the compact set MpL , q. The limit construction of Pp q from is translated to the operation of taking holomorphic envelope.

Proposition 4.1.7 (2.3.4). -In CpL q an ,o n eh a s { M ´p q"Mp q.

Our study of the continuity problem is based on geometric considerations of the dual unit disc bundles. The point of attack is the effect of dilation along fibers of pL _ q an to the dual unit disc bundle with respect to a metric. Recall that for any ✏ P R, we denote by p✏q the dilated metric defined by @x P X an , @s n P R n pL q, |s n pxq| n p✏q :" |s n pxq| n ¨en✏

Note that for ✏ °0, the dilated metric is bigger, so D _ pL , q is contained in D _ pL , p✏qq.

The metric is upper/lower-semicontinuous if the function x fi Ñ |s n pxq| n is so. The observation is that for an upper semicontinuous metric, the dilation on its dual unit disc bundle detects the possible discontinuity of the metric. _ pL , p✏qq (resp. M ´p p✏qq) is a neighbourhood in VpL q an of D _ pL , q (resp. neighbourhood in CpL q an of M ´p q)f o ra n ys u fficiently small ✏ P R °0. In other words, the metric is continuous if and only if D _ pL , q is contained in the (topological) interior of D _ pL , p✏qq.

With this geometric criterion, the continuity properties of and Pp q are related by the following diagram. Each term is seen as a topological space with the subspace topology inherited from the ambient space VpLq an (resp. CpLq an ). The symbol î means inclusion in the topological interior, and x p¨q means the operation of taking holomorphic convex envelope in the ambient space. O O

Equivalently one may look at the following diagram, where one sends the previous diagram by the map p an : VpL q an Ñ CpL q an M ´p p✏qq x ¨/ / Mp p✏qq M ´p q

x ¨/ / p?q]

O O

Mp q p?q]

O O

To transfer the continuity of to the continuity of Pp q, it then suffices to show the "commutativity" of this diagram of operations. In fact such a phenomenon of "commutativity of holomorphic convex envelope and neighbourhood" should hold in a more general setting, for any compact set inside any ambient space of affine analytic space. This commutativity is indeed the content of Speculation 4.2.1.

With this commutativity, one easily deduces the continuity of Pp q from the continuity of .

Note that our method is of global nature, and the ampleness assumption on L is thus important.

Holomorphic convexity of holomorphic envelope

In this section we make a synopsis of the following commutativity property, listing some attempts and difficulties, without being able to give a complete proof. This section does not contain any result and can be jumped. Speculation 4.2.1.-Assume that pk, |¨|q is spherically complete. Let Z be the analytification of a normal affine variety. For any compact set K and U in Z with K î U , one has p K î p U . In other words, for and any neighbourhood U of K in Z, the envelope p U is a neighbourhood of the envelope p K. (The holomorphic envelope is taken in Z.) 4.2.1. Reformulation by holomorphic convexity. -Indeed, one can reformulate the commutativity property as confirming the holomorphic convexity of p U (in Z). Namely, any relative compact subset K contained in Int t p U has a holomorphic envelope (in Z) p K which should be still contained in Int t p U . The notion of Stein spaces and holomorphic convexity in the Berkovich non-Archimedean analytic setting has been recently systematically studied in [START_REF] Maculan | Notions of Stein spaces in non-archimedean geometry[END_REF]. In particular, analogues of important classical results concerning Stein spaces has been established, such as Cartan theorem A and B, as well as many other fundamental intermediate results. As coherent analytic sheaf methods are used, mainly in their study, holomorphic convexity of open sets are dealt with.

Here for our application to the continuity problem of envelope metrics, we need to establish the holomorphic convexity for the holomorphic envelope of a closed set, namely Mp p✏qq in our case. As we do not know much about the boundary and interior of this closed envelope set, it seems not easy to apply directly results from [START_REF] Maculan | Notions of Stein spaces in non-archimedean geometry[END_REF] to establish holomorphic convexity. Hence we perform another study. 4.2.2. Core issues. -To show the key result of holomorphic convexity, for any z 0 P p K, one needs to find a neighbourhood V of z 0 in Z, itself contained in p U . There are two basic issues to be dealt with: 1. To characterize points such as z 0 , in holomorphic envelope p U , by knowledge of the original set U ; 2. To propagate such characterizations to some neighbourhood around z 0 P p K, from the assumption K î U . In other words, performing an "analytic continuation" of the characterization.

Inspired by the approaches in the complex analytic setting, one has basically two different points of view. One shall see that each of them individually encounters obstacles in the non-Archimedean setting, yet a combination of them might possibly work.

Note that in our case holomorphic envelope coincides with the polynomial envelope. Thus it suffices to use polynomials to define the envelope set. This strategy is inspired by the Cartan-Thullen's argument in the C-analytic setting. For simplicity of presentation one may look at the case where Z is pA n q an . Let ~¨~K (resp. ~¨~U )b et h es u pn o r mo nK (resp. on U ). For any polynomial f P krT T T s, one can try to do the Taylor expansion at z, and gets

f pwq" ÿ LPN d B L f pzq¨pT T T pwq´T T T pzqq L ,
where B L f pzq are multi-partial derivatives of f pzq (Taylor coefficients) which are still polynomials. Then the coefficients |B L f pzq| are controlled from above by ~BL f ~K thanks to the fact that z P p K. Moreover, as K î U , by Cauchy estimates one can find °0 such that each term of the form ~BL f ~K ¨ |L| is bounded from above by ~f ~U . Thus for points w in the neighbourhood of z in Z defined by the inequalities |T i pwq´T i pzq| † , one has |f pzq| § ~f ~U . Thus this neighbourhood is contained in p U . Besides, one may also reformulate the above argument by considering the action of an analytic group pG n a q an on pA n q an . The theory developped in [START_REF] Vladimir | Spectral theory and analytic geometry over non-Archimedean fields[END_REF]Chapter 5] implies, by the continuity of the action, that the convolution point ⇠p q˚z with a Gauss point ⇠p q is contained in p U if is small. Thus the set { t⇠p q˚zu which is just the polydisc neighbourhood of z constructed by Taylor expansion, is also contained in it.

The central difficulty to carrying out this argument in the non-Archimedean setting is the possibility of performing Taylor development at z. In general, the point z is not a rigid point of Z, this fact impeaches a Taylor development. If one uses base field change to a larger valued field pk 1 , |¨| 1 q to make z a k 1 -rigid point in Z k 1 , the holomorphic envelope behaves well with respect to base change thanks to the study in [START_REF] Maculan | Notions of Stein spaces in non-archimedean geometry[END_REF]. But the obtained neighbourhood on ⇡ ´1 k 1 {k pZq does not necessarily descend to a neighbourhood on Z,a st h e continuous surjective map ⇡ ´1 k 1 {k : Z k 1 Ñ Z is not open in general, especially when the field extension is large. In terms of the reformulation of action by pG n a q an , the convolution point ⇠p q˚z may just coincide with z itself in case where z is itself a "big" point and is small (for example, one has ⇠p q˚⇠p1q"⇠p1q). 4.2.4. Spectral-theoretic point of view. -From the spectral-theoretic point of view, one views the holomorphic envelope of a set as the analytic spectrum of the Banach algebra of holomorphic functions defined on this set. Points in the holomorphic envelope are characterized by the continuity of the associated evaluation homomorphism from this stuctural Banach algebra, and the analytic continuation is realized thanks to the perturbation stability of index of Fredholm operators.

More precisely, one takes again Z to be pA n q an for illustration of ideas. One forms the k-Banach algebra BpU q as the completion of polynomials krT T T s with respect to the sup norm ~¨~U .A ta n yp o i n tz P Z, one may look at the evaluation homomorphism evpzq from (the dense subalgebra of polynomials of) BpU q to p pzq.P o i n t sz in p U are characterized by the continuity (i.e. boundedness) of evpzq. In addition, in view of the Cauchy estimates, points z in the interior of p U should be characterized by the Fredholmness of the evaluation homomorphism evpzq, a property stronger than just being continuous. Indeed, since z 0 P p K, the relative compacity K î U implies that the evaluation homomorphism evpz 0 q is completely continuous, hence is Fredholm. Moreover, the index of evpz 0 q should be positive. As small perturbations of Fredholm operators are still Fredholm with the same index, one sees that if z is in a sufficiently small neighbourhood of z 0 P p K, the evaluation homomorphism evpzq is still Fredholm with positive index. Thus this neighbourhood is in the interior of p U , hence is also in p U . There are multiple difficulties to carrying out this argument in non-Archimedean setting. The first one is again about the rigidness of point z 0 . Contrary to the C-analytic case, the evaluation morphisms evpzq lands in different target Banach spaces p pzq.A l t h o u g h perturbation theory of Fredholm operators is well-developped in [START_REF] Serre | Endomorphismes complètement continus des espaces de Banach p-adiques[END_REF] for fixed source and target Banach spaces, one does not know how to formulate perturbation theory for varying target Banach space. A second difficulty is whether the Fredholm property (with non-zero index) of evpzq really characterizes points of the interior of spectrum p U .O n ew o u l dl i k e to use the Koszul complex of BpU q with respect to a regular sequence tT i ´Ti pzqu iPt1,...,du to study the properties of evpzq. The situation is not so clear even in the C-analytic case (except in dimension 1).

4.2.5. Mixed point of view. -In view of difficulties encountered in each of the above cases, it seems plausible to treat them separately, according to the nature of the point z. On one extremity, if z is "small", namely a rigid point, one can produce a desired neighbourhood by the first method. On another extremity, if z is "large", for example a point with p z being t0u, or even more, an Abhyankar point, then one should use a method of different nature to produce the desired neighbourhood. It seems that variabilities of "small" points are geometrical while variabilities of "large" points are spectral. In the intermediate case one can decompose the variability of z into two types of directions along which z behaves as in the two extreme case. We leave this study for the future.

Neighbourhood of dual unit disc bundle and continuity of metric

One first gives a criteria of continuity of an upper-semicontinuous metric in terms of topological property of its dual unit disc bundle. _ pL , p✏qq is a neighbourhood of D _ pL , q for any sufficiently small ✏ °0.

Proof.-The criterion is clear from the following geometric picture, while the argument is somehow cumbersome. It's obvious that if is continuous, then the dilated dual unit disc bundle is a neighbourhood of the original dual unit disc bundle. Conversely, if is not continous, let x 0 P X an be one of the discontinuity point. Take any global section s n P R n pL q such that s n px 0 q‰0. By the discontinuity of at x 0 ,o n e has lim inf Denote by R the subset of X an of the points x satisfying |s n pxq| n § e n {2 ¨lim inf Since x 0 is a discontinuous point, by construction, for any neighbourhood U of x 0 in X an , one has U X R ‰H.

For any ✏ P R such that 0 † ✏ † {4, we show that any open neighbourhood Q of ⇠ 0 x 0 in VpL q an can not be contained in D Since the map ⇡ an : VpL q an Ñ X an is open, the set ⇡ an pQq is an open neighbourhood of x 0 , hence has non-empty intersection with R. Let x be a point in R X ⇡ an pQq.F o ra n y z Pp⇡ an q ´1pxqXQ, by the construction of Q one has x is not contained in Q. Thus Q is not contained in D _ pL , p✏qq.F r o mt h i s ,o n e deduces that D _ pL , p✏qq is not a neighbourhood of D _ pL , q.

Hence the criteria for continuity by neighbourhoodness holds.

Recall that for any upper-semincontinuous metric , we denote by M ´p q the compact set p an pD _ pL , qq and by by Mp q the compact set p an pD _ pL , Pp qqq in the affine cone CpL q an . Corollary 4.3.2.-Let be an upper-semicontinuous metric on L an . It is continuous if and only if M ´p p✏qq is a neighbourhood of M ´p q for any sufficiently small ✏ °0. Dual unit disc bundle of pL , q and its dilation by e ✏ in VpL q an Theorem 4.3.3.-Suppose that pk, |¨|q is spherically complete. Suppose that X is normal. Assume that Speculation 4.2.1 holds. If is continuous, then Pp q is also continuous.

Remark 1 .

 1 2.5.-A basis of the Berkovich topology is given by basic open sets, which are sets of the form U pf ; p, qq :"tz P MpAq : p † |f | z † qu indexed by pp, qqPR 2 and f P A. A general open set is thus a union of finite intersections of basic open sets. Proposition 1.2.6.-Let A be a k-Banach algebra. Then MpAq is a non-empty compact Hausdorff topological space. ([3, Theorem 1.2.1])

Proposition 1 . 2 .

 12 14.-Let A be a k-Banach algebra. For any f P A,o n eh a s( [ 3, Theorem 1.3.1]) ~f ~sp " max zPMpAq |f |pzq. Definition 1.2.15.-Let A "p A, ~¨~q be a Banach k-algebra. The radical of A is defined as the null-space of its spectral seminorm ~¨~s p , denoted by radpAq or by radp~¨~q. Note that radpAq" f P A, lim nÑ`8

Remark 1 . 2 . 1 Ñ

 121 21.-Suppose that the absolute value |¨| on k is non-trivial. If a k-Banach algebra homomorphism is finite as homomorphism of k-algebra, and A 1 is Noetherian, then is automatically Banach finite: there is a surjective A 1 -module homomorphism p : A 'n A 2 ,b yP r o p o s i t i o n1.2.19 kerppq is closed. Then p is continuous hence is admissible by Corollary 1.1.5.S oA 2 is a Banach finite A 1 -module.

Remark 1 .

 1 3.4.-An affinoid algebra norm and the quotient algebra norm of Gauss algebra norm by the defining admissible surjective homomorphism are just equivalent (see Theorem 1.1.4) but not necessarily equal. Example 1.3.5.-The quotient Banach algebra of an affinoid algebra is an affinoid algebra. Proposition 1.3.6.-Suppose that the absolute value |¨| on k is non-trivial. Let C be a k-Banach algebra which is finite over an affinoid algebra A, then C itself is an affinoid algebra. ([8,P r o p o s i t i o n6 . 1 . 1 . 5 ] ) Proposition 1.3.7.-Let B be a Banach k-algebra with its Banach algebra norm ~¨~.

Proposition 1 . 3 .

 13 21.-Let A be a reduced strict affinoid algebra. Then its spectral norm ~¨~A ,sp is a complete norm on A. It is equivalent to the Banach algebra norm ~¨~A. ([29, Theorem 3.4.9], [8, Theorem 6.2.4.1]) Corollary 1.3.22.-Let A be a reduced general affinoid algebra. Then there exists C °0 such that ~f ~ § C~f ~sp for all f P A. In particular, ~¨~s p is complete on A ,a n d is equivalent to ~¨~.( [ 3,P r o p o s i t i o n2 . 1 . 4 . i i ] ) 1.3.4. Affinoid space as locally ringed space. -The Berkovich spectrum of affinoid algebras are called affinoid spaces. It is possible to put locally ringed space structures on them. The construction of structural sheaf goes first with a Grothendieck topology generated by closed compact subsets of affinoid domains, then passes to the canonical topology by a limit process approximating an open set by these compact sets. 1.3.4.1. Affinoid domains and structural algebra.-Definition 1.3.23.

Theorem 1 . 3 .

 13 34.-Let A be a strict affinoid algebra and V an admissible covering by strict affinoid domains for MpAq. Then the differentials d l have closed images and the complex C ' pA, Vq is exact. ([8,P r o p o s i t i o n8 . 2 . 2 . 5 ] ) Corollary 1.3.35.-For general affinoid domain MpAq with general affinoid domains covering V, the complex C ' pA, Vq is exact where the differentials have closed images. So is C ' pM, Vq for any finite Banach A-module M .( [ 3,P r o p o s i t i o n2 . 2 . 5 ] ) .

Proposition 1 . 3 .

 13 39.-O MpAq is a sheaf of local rings. The topological space MpAq has a structure of locally ringed space given by the sheaf O MpAq .( [ 3, Section 2.3]) 1.3.4.3. Change of base valued field. -Let A be a k-affinoid algebra and X be the affinoid space MpAq. Let pk 1 , |¨| 1 q be a ultrametric valued field extension of pk, |¨|q. Denote by A 1 the k 1 -affinoid algebra A p b k k 1 , it has also a structure of k-Banach algebra. Denote by i k 1 {k : A Ñ A 1 the canonical homomorphism of k-Banach algebras. Definition 1.3.40.

Definition 1 .

 1 4.1.-Let A and B be Banach k-algebras, and : A Ñ B be a homomorphism of Banach algebras. We call spectrum of homomorphism and we denote by Σ the image of MpBq in MpAq under ‹ . Definition 1.4.2.-Let A be a Banach k-algebra. Let K be a compact subset of MpAq.

  Lemma 1.4.3.-The intersection of all Weierstrass neighbourhoods of K in MpAq coincide with p K.( [ 3,P r o p o s i t i o n2 . 6 . 1 ] ) Proposition 1.4.4.-Let A be a k-affinoid algebra, B be a Banach k-algebra. Let : A Ñ B be a homomorphism of Banach k-algebras. Let B 1 be the closed sub-algebra generated by the image of of A in B and let 1 : A Ñ B 1 be the restricted homomorphism. Then x Σ " Σ 1 .( [ 3,P r o p o s i t i o n7 . 3 . 1 ] ) Corollary 1.4.5.-Let : A Ñ B be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra with dense image. Then Σ is holomorphically convex. One has the following analogue of Arens-Calderon theorem, which holomorphically convexifies the spectrum of a homomorphism of Banach k-algebras by adding variables on the source algebra. Proposition 1.4.6.-Let A be a k-affinoid algebra, B be a Banach k-algebra. Let : A Ñ B be a homomorphism of Banach k-algebras. Then for any open neighbourhood U in MpAq of the spectrum Σ , there exists a homomorphism of Banach algebras extending

Lemma 1 .

 1 4.7.-Let : A Ñ B be a Banach algebra homomorphism from an affinoid algebra A to a Banach algebra B. Then for any Laurent domain neighbourhood V of Σ , extends to a unique Banach algebra homomorphism V : A V Ñ B.( [ 3, Corollary 2.5.16]) Theorem 1.4.8.-Let : A Ñ B be a homomorphism of Banach algebras from an affinoid algebra to a Banach algebra. Let V Ñ MpAq be any special domain containing Σ . Then there exists a Banach algebra homomorphism

Proposition 1 .

 1 5.10.-Let : X Ñ Y be a morphism of schemes of locally finite type over Spec k. Then it induces a continuous map an : X an Ñ Y an .A n d is (1) separated, (2) injective, (3) surjective, (4) an open immersion and (5) an isomorphism if and only if an has the same property. ([3,P r o p o s i t i o n3 . 4 . 6 ] ) Theorem 1.5.11.-If X is proper, then X an is Hausdorff and compact. ([3, Theorem 3.4.8])

2. 1 . 1 . 4 .

 114 Fiber of total space.-F o ra n yx P X, we consider the base change morphism Spec pxqÑX. We denote by L pxq the invertible sheaf L b O X pxq over Spec pxq, which can be seen as a rank one pxq-free module. The fiber product VpL qb O X pxq is an affine pxq-scheme whose coordinate ring is the graded pxq-algebra Sym pxq L pxq.W e denote by VpL qpxq this pxq-scheme and by ⇡ L pxq : VpL qpxqÑSpec pxq the structural morphism, which inscribes into a cartesian diagram of morphism of k-schemesVpL qpxqFor any x P X, we shall denote by R ' pL qpxq the graded pxq-algebra R ' pL qb k pxq.I f U is an affine open subset of X which contains x, then the restriction morphism of graded k-algebras R ' pL qÑ à nPN H 0 pU, L bn q induces a morphism of pxq-algebras L pxq : R ' pL qpxq ›Ñ Sym pxq L pxq. Similarly, in considering L | Y , for any y P Y , we denote by L | Y pyq the corresponding morphism of graded pyq-algebras. For any e P L pxqzt0u, one can identify the graded pxq-algebra Sym pxq L pxq with the one-variate polynomial algebra pxqrT s by identifying e with T .

Proposition 2 .

 2 1.4.-Let R ' be a graded k-algebra. If ~¨~is an ultrametric algebra seminorm (see Definition 1.2.1)o nR ' , then the map ~¨~`: R ' Ñ R •0 , defined as ~ps n q nPN ~`:" sup nPN ~sn ~,

  r n " ÿ mPt0,...,nu s m ¨tn´m . If the family pk¨kq nPN is sub-multiplicative, by the ultrametricity assumption on k¨k n ,o n e obtains that kr n k n § max mPt0,...,nu ks m ¨tn´m k n § max mPt0,...,nu ks m k m ¨kt n´m k n´m , so ~r~is bounded from above by ~s~¨~t~.

Lemma 2 . 1 .

 21 12.-If ~¨~is ultrametric, then ~¨~s p is also ultrametric.Proof.-For any pa, bqPA 2 , by the definition of spectral seminorm, for any ✏ °0, there exists C ✏ P R °0 such that for any n P N one has ~an ~ § C ✏ ¨p~a~s p `✏q n , ~bn ~ § C ✏ ¨p~b~s p `✏q n .

Lemma 2 . 1 .

 21 14.-Let R ' " À nPN R n be a graded k-algebra and ~¨~be a graded algebra seminorm on R ' . Let |¨| z be an ultrametric multiplicative seminorm on R ' . Then |¨| z is also an ultrametric multiplicative seminorm. If |¨| z is bounded from above by ~¨~,so is |¨| z .I f|¨| z is a point in Mp p R ' , ~¨~q, then |¨| z corresponds also to a point in Mp p R ' , ~¨~q.

  Proof.-Denote by k¨k n ( nPN the family of norms of restrictions of ~¨~on S n ( nPN .O n e can construct a family of seminorms k¨k 1 n ( nPN on R n ( nPN as follows.

Corollary 2 . 1 .

 21 19.-With the same hypothesis and notations as above, let |¨| z be an ultrametric power-multiplicative seminorm on R ' .D e n o t es t i l lb y|¨| z its restriction on S ' . Then |¨| z § ~¨~on S ' if and only if |T | z § ~T ~1,i fa n do n l yi f|¨| z § ~¨~1 on R ' , Proof.-For the direct implication, if |¨| z § ~¨~on S ' ,t h e no n eh a s|T n 0 | z § ~T n 0 ~, so |T | z § ~T ~1. Since both seminorms are ultrametric and power-multiplicative, one has |¨| z § ~¨~1 on R ' . The inverse implications are easy. 2.1.5. Analytic spaces. -2.1.5.1. Analytification.

  xPX an |s m pxq| m ¨|s n pxq| n § sup xPX an |s m pxq| m ¨sup xPX an |s n pxq| n " ks m k m ¨ks n k n and ks l n k ln " sup xPX an |s l n pxq| ln "p sup xPX an

Remark 2 .

 2 1.27.-The reducedness of closed sub-scheme Y is necessary for the semisimplicity of the Banach k-algebra p R ' pL X|Y , X|Y q.

2. 3 . 4 . 1 .

 341 From outside. -We embed the Berkovich spectrum into the analytification of the affine cone.

Fact 2 .

 2 3.5.-Let S ' pL q be a graded algebra, and let ~¨~be an algebra norm on S ' pL q. The inclusion S ' pL qÑ p SpL , ~¨~q induces a continuous map p SpL , ~¨~q Ñ CpL S' q which is homeomorphism from domain to image. Moreover, one can find an affinoid algebra norm on S ' pL q, denoted by ~¨~3,s u c ht h a t~¨~ § ~¨~3.A n do n eh a sah o m o m o r p h i s m of Banach algebras p SpL , ~¨~3qÑ p SpL , ~¨~q, the induced continuous map Mp p SpL , ~¨~qq Ñ Mp p SpL , ~¨~q 3 q

2 .

 2 Localization of Banach algebra homomorphism. -We shall use holomorphic functional calculus to construct Banach algebra homomorphisms. Recall that for any homomorphism of Banach algebras f : A Ñ B, the spectrum Σ f of the homomorphism is the image of MpBq in MpAq under the map f ˚. The theory of functional calculus (Theorem 1.4.8) provides a factorization of f through a localization: assume that A is affinoid, for any special domain W containing Σ f , let W be the structural Banach algebra of W , and denote by i W is the localization homomorphism corresponding to the inclusion W ãÑ MpAq, then there exists a Banach algebra homomorphism ✓ which makes the following diagram commutative

3. 4 . 1 .

 41 Algebra norm induced by Fubini-Study metric. -3.4.1.1. Case for pP d , Op1qq.

Proposition 3 .

 3 4.1.-For any J P N d`1 ,o n eh a s kT T T J k |J| " π iPt0,...,du

Proposition 3 .

 3 4.3.-Assume that kT i k ( iPt0,...,du are pk, |¨|q-free. Let S Ñ N d`1 be a finite set of multi-indices, then for any J P S any J

/

  / / / CpL q an

Proposition 4 . 1 . 8

 418 (4.3.1). -Let be an upper-semicontinuous metric on L . The metric is continuous if and only if D

D

  _ pL , p✏qq x ¨/ / D _ pL , Pp qp✏qq D _ pL , q x ¨/ / p?q] O O D _ pL , Pp qq p?q]

4. 2 . 3 .

 23 Function-theoretic point of view. -From the function-theoretic point of view, points in a holomorphic envelope set, such as z, are detected by the defining boundedness conditions of the corresponding evaluation map, and the analytic continuation is simply based on the Taylor development of the function around z.

Proposition 4 .

 4 3.1.-Let be an upper-semicontinuous metric on L . The metric is continuous if and only if D

  xÑx 0 |s n pxq| n † lim sup xÑx 0 |s n pxq| n . As the metric of is upper-semicontinous, one has lim sup xÑx 0 |s n pxq| n § |s n px 0 q| n . For any ✏ P R,a n da n yx P X an , let ⇠ x p✏q be the Gauss point of the dual unit disc over x in D _ pL , p✏qq.O n eh a s |s n pxq| n p✏q " |s n | ⇠xp✏q . The upper-semicontinuity of the metric of implies that there exist °0 such that lim inf xÑx 0 |s n pxq| n " |s n | ⇠x 0 p q .

x 1

 1 Ñx 0 |s n px 1 q| n , or equivalently the points x satisfying |s n | ⇠xp0q § e n {2 ¨|s n | ⇠x 0 p´ q " |s n | ⇠x 0 p´ {2q .

  _ pL , p✏qq.O n em a ya l s oa s s u m et h a tQ is contained in the open set tz : |s n | z °|s n | ⇠x 0 p´ {4q u up to replacing Q by its intersection with this open set. See the picture blow.

  |s n | z °|s n | ⇠x 0 p´ {4q , whereas by the construction of R and the choice of ✏ one has |s n | ⇠xp✏q " e n✏ ¨|s n | ⇠xp0q † e n✏ ¨|s n | ⇠x 0 p´ {2q † |s n | ⇠x 0 p´ {4q , so ⇠ ✏

x 0 ⇠ x 0

 00 p´ q⇠ x 0 p0q ⇠ x 0 p´ {2q x ⇠ x p✏q ⇠ x p0q |s n | z § |s| ⇠x 0 p´ {2q |s n | z " |s| ⇠x 0 p´ {4q Q VpL q an

  norme d'algèbre sup ~¨~ | Y . Ce couple sup induit aussi une métrique enveloppante Pp | Y q sur L | Y . Le résultat en haut implique que les deux soient égales. Théorème 0.2.3 (Proposition 2.4.3). -Pour toute norme d'algèbre graduée ~¨~sur R ' pL q, on a l'égalité p~¨~X |Y q sp " ~¨~Y .

Ainsi, si est une métrique semipositive, alors Pp X|Y q" | Y . 0.2.2. Chapitre 3. -Extension de sections du fibré restreint avec un contrôle de norme 0.2.2.1. Problème et ses reformulations. -On étudie un problème d'extension de sections d'un fibré en droites semi-positivement métrisé pL , q restreint à une sous-variété fermée. Étant donné un entier n P N, une sous-variété fermée Y de X, et une section t n de L bn | Y , on cherche une section globale s n de X dont la restriction à Y s'identifie à t n et dont la norme sup est contrôlé en fonction de n et de la norme de t n . L'existence d'une telle section dont la norme sup «se concentre autour de sous-variété» est très utile pour étudier le volume du fibré en droites avec un argument inductive en dimension.

  2.3,l e domain spécial W ✏ peut être construit de telle sorte qu'il soit contenue dans la partie compacte Mp p R Méthode d'approximation algébrique. -Cette méthode tente de comparer directement la norme d'algèbre de Banach ~¨~ X|Y avec une norme d'algèbre affinoïd. D'abord on montre que la norme d'algèbre ~¨~ X|Y est affinoïd sous l'hypothèse que soit elle-même une métrique Fubini-Study. Avec une hypothèse supplémentaire que la valeur absolue |¨| sur k soit discrète, on réussit à calculer explicitement la norme ~¨~ dans le cas où X est l'espace projective, L est le fibré Op1q, et est une métrique Fubini-Study de forme FSpk¨k 1 q pour une norme ultramétrique k¨k 1 sur R 1 pL q. Dans ce cas particulier, l'algèbre de section est isomorphe à l'algèbre symétrique sur l'espace vectoriel R 1 pL q. En exploitant au fond de l'existence d'une base orthogonale non archimédienne pour pR 1 pL q, k¨k 1 q, on montre que la norme d'algèbre ~¨~ est une norme de Gauss de multi-rayons donnée par les normes de cette base orthogonale. Il s'en suit que quand est une métrique Fubini-Study de forme FSpk¨k 1 q, la norme d'algèbre ~¨~ X|Y est affinoïd pour X, Y générales et L très ample. Par conséquence on a la majoration désirée dans

' pL X|Y , | Y p✏qqq. L'inclusion spectrale W ✏ ãÑ Mp p R ' pL X|Y , | Y p✏qqq implique la majoration de ~¨~W ✏ par ~¨~ | Y p✏q . La combinaison de ces deux majoration implique la majoration désirée. 0.2.2.3. ce cas. Si on peut généraliser cette propriété d'affinoïdité de la norme d'algèbre pour L ample et est une métrique n-ième Fubini-Study (de forme 1 n FSpk¨k n q), alors on saura traiter le cas où la métrique semi-positive est générale par l'approximation uniforme de par les métriques n-ième Fubini-Study pour n grand. 0.2.3. Chapitre 4. -Continuité de la métrique enveloppante 0.2.3.1. Problème de continuité de la métrique enveloppante. -On étudie une propriété de régularité de la métrique enveloppante. Comme la métrique Pp q est une limite ponctuel le de métriques (Fubini-Study) continues, elle n'est pas nécessairement continue.

  0.2.3.2. Effet de régularisation: voisinage et convexité holomorphe.-O n p r o p o s e un argument géométrique pour étudier la régularité de Pp q,s u i v a n tu np o i n td ev u e «Berkovich». Notre approche est purement analytique et donc indépendante de modèles entiers. Théorème 0.2.7 (Theorem 4.3.3). -Supposons que la Speculation 0.2.8 soit vraie. Si est continue, alors Pp q l'est aussi. L'argument est basé sur l'observation suivante. Rappelons que pour une métrique qui est semi-continue supérieurement, on regard son fibre du disc unite dual D

_ pL , q dans

  1.8.-The norm k¨k _ is ultrametric, and k¨k __ " k¨k if and only if k¨k is ultrametric. ([24, Section 2.2.3])

	Definition 1.1.9.

  Lemma 1.1.11.-Let f : V Ñ W be a surjective k-linear map of finite-dimensional vector spaces, with dim k W " 1. Let k¨k V be a norm on V and let k¨k W be its quotient norm by f . Then the norm k¨k W,k 1 identifies with the quotient norm of k¨k V,k 1 induced by the surjectivek 1 -linear map f b id k 1 : V k 1 Ñ W k 1 .( [ 24, Lemma 2.5])Moreover, it is said to be orthonormal if in addition ke i k " 1 for all i Pt1,...,nu.Ab a s i s te i u iPt1,...,nu of V is called ⇢-orthogonal (for some ⇢ P R •1 )i f Lemma 1.1.13.-Let pV, k¨kq be a finite-dimensional ultrametrically normed vector space over k.I ftv i u iPt1,...,nu is a finite set of elements of V such that tkv i ku iPt1,...,nu are distinct in R `. Then k ∞ iPt1,...,nu v i k " max iPt1,...,nu kv i k. Proof.-If n " 2, this is clear from the ultrametric inequality. The general case then follows by induction on n.Corollary 1.1.14.-Let pV, k¨k V q be a finite-dimensional ultrametrically normed vector space over k.S u p p o s et h a tpk, |¨|q is discretely valued. If te i u iPt1,...,nu is a basis of V such that tke i ku iPt1,...,nu are pk, |¨|q-free, then te i u iPt1,...,nu is an orthogonal basis.Proof.-For any f "p f 1 ,...,f n qPp k ˆqn , the numbers t|f i | ¨ke i ku iPt1,...,ru are distinct, otherwise there exist i, j Pt1,...,nu,i ‰ j such that

	@pc 1 ,...,c r qPk n , ⇢	¨› › ›	ÿ iPt1,...,nu	c i e i	› › › • max iPt1,...,nu	kc i e i k
	logke i k ´logke j k " log	f i f j	P log|k	ˆ|
	which contradicts the assumption of their Q-freeness for pk, |¨|q. Hence by Lemma 1.1.13 ÿ iPt1,...,nu f i e i " max 0 §i §n kf i ¨ei k.
	Definition 1.1.15.-A valued field pk, |¨|q is said to be spherically complete,i ft h e
	intersection of every decreasing sequence of balls Bp n ,r n q"t P k, | ´ n | § r n u is
	non-empty.
	Proposition 1.1.16.-Let pV, k¨k V q be a finite-dimensional ultrametrically normed
	vector space over k.I f|¨| is non-trivial and pk, |¨|q is spherically complete, then pV, k¨k V q
	admits an orthogonal basis ([8,P r o p o s i t i o n2 . 4 . 4 . 2 ] ) .
	Remark 1.1.10.-If k¨k is ultrametric, then k¨k k 1 is the largest ultrametric norm on Proposition 1.1.17.-Let pV, k¨k V q be a finite-dimensional ultrametrically normed V k 1 extending k¨k.( [ 24, Definition 2.4]) vector space over k.I f |¨| is non-trivial, then for any ⇢ P R •1 , pV, k¨k V q admits a ⇢-
	orthogonal basis ([8,P r o p o s i t i o n2 . 6 . 1 . 1 ] ) .
	1.2. Banach algebra
	1.2.1. Basic constructions. -
	1.1.2. Orthogonal basis. -Definition 1.2.1.
	Definition 1.1.12.-Let pV, k¨kq be a finite-dimensional normed vector space over k.
	Ab a s i ste i u iPt1,...,nu of V is called orthogonal (with respect to k¨k)i f
	@pc 1 ,...,c r qPk n ,	› › ›	ÿ iPt1,...,nu	c i e i	› › › " max iPt1,...,nu	kc i e i k.

  MpA 2 qÑMpA 1 q the map sending any element z P MpA 2 q to the point corresponding to the seminorm | p¨q| z , called the map associated with the homomorphism of k-Banach algebras . The map ‹ is continuous (see [3, Remark 1.2.2 (iii)]). Proposition 1.2.11.-If : A 1 Ñ A 2 is a homomorphism of Banach k-algebras with dense image, then ‹ is an injective map whose image is closed.Proof.-The map ‹ is injective, since for any two points z 1 and z 2 in MpA 2 q whose image under ‹ are the same, the restriction of |¨| 1 and |¨| 2 on the image of are equal, hence the two multiplicative seminorms on A 1 | 1 p¨q| 1 and | 2 p¨q| 2 are equal by the density of image.

	Definition 1.2.8.-Let A be a Banach k-algebra. For any f P A,theGelfand transform
	of f is defined as the element p f :"pf pzqq zPMpAq in π pzq.
	zPMpAq
	Proposition 1.2.9.-Let A be a Banach k-algebra. An element f P A is invertible if
	and only if f pzq‰0 for any z P MpAq ([3, Corollary 1.2.4]).
	1.2.3. Continuous map. -
	Definition 1.2.10.-Let : A 1 Ñ A 2 be a homomorphism of Banach k-algebras. One
	denotes by ‹ :

  [8, Theorem 5.2.1.2], [29, Theorem 3.1.1]) Proposition 1.3.14.-The Tate algebra T n is Noetherian. All of its ideals are closed. For strict affinoid algebra A, there exists an injective and admissible Banach algebra homomorphism T d Ñ A for some d °0, which endows A with a structure of finite algebra over T d . Moreover, d equals the Krull dimension of A.( [ 8, Theorem 6.1.2.1], [29, Theorem 3.2.1]) Corollary 1.3.17.-Let m be a maximal ideal of strict affinoid algebra A, then A{m is a finite extension of k.

([8, Theorem 5.2.6.1, Corollary 5.2.7.2], [29, Theorem 3.2.1]) Corollary 1.3.15.-Any affinoid algebra is Noetherian. All of its ideals are closed ([3, Propositon 2.1.3]). Proposition 1.3.16 (Noether normalization). -Proposition 1.3.18.-Let : T d Ñ B be an integral torsion-free homomorphism of Banach algebras. Then for each b P B,o n eh a s ~b~B ,sp " max 1 §i §n ~ai ~1{i T d ,sp , where P pT q"T n `a1 X n´1 `¨¨¨`a b P ArT s is the minimal polynomial of b over T d .( [ 8, Proposition 3.8.2.5]) Corollary 1.3.19.-Let : A Ñ B be an integral homomorphism of affinoid algebras. Then the same formula holds as in this Proposition. ([8,P r o p o s i t i o n6 . 2 . 2 . 4 ] ) 1.3.3. Topological structures: the spectral norm.

  Definition 1.3.30.-The union of any finite family of affinoid domains in MpAq is called a special domain in MpAq. Definition 1.3.31.-The Grothendieck topology on MpAq is the one with special domains as admissible open sets and finite coverings of special domain as admissible coverings. One denotes by MpAq G the space equipped with this G-topology. Remark 1.3.32.-Indeed, by Proposition 1.3.28 and Proposition 1.3.25,theadmissible coverings defined here verifies the required axioms.

	Definition 1.3.33.

Proposition 1.3.28.-A finite intersection of affinoid domains is an affinoid domain. ([3, Remark 2.2.2.iv]) Corollary 1.3.29.-Any point z P MpAq has a fundamental system of (closed) neighbourhoods consisting of affinoid domains. ([3,P r o p o s i t i o n2 . 2 . 3 ] ) 1.3.4.2. Special domains and acyclicity of structural presheaf.-

  The structural pre-sheaf of affinoid algebras O MpAq G on MpAq G (with respect to the G-topology)i st h eo n e assigning V the k-Banach algebra A V . It is a sheaf thanks to Corollary 1.3.35. Remark 1.3.37.-The k-Banach algebra O MpAq G pV q does not depend on the way of being a union of affinoid domains. Definition 1.3.38.-For any open subset U of MpAq, let O MpAq be the pre-sheaf of k-algebras (with respect to the canonical topology) which assigns U the limit

  Moreover, the restriction of p L to VpL qzO L pXq is an open immersion, whose image is contained in CpL qz0 0 0 L .I fX is proper, then p L restricted to VpL qzO L pXq defines an isomorphism between VpL qzO L pXq and CpL qz0 0 0 L .( [ 33, Proposition 8.8.2, Remark 8.8.3])

  2.1.3. Completion. -Let R ' " À nPN R n be a graded k-algebra and ~¨~be an ultrametric seminorm on R ' . Denote by p Rp~¨~q the separated completion of À nPN R n with respect to ~¨~. The seminorm ~¨~on À nPN R n induces by continuity an ultrametric norm on p Rp~¨~q which we still denote by ~¨~by abuse of notation. Proposition 2.1.11.-If ~¨~is an ultrametric and graded norm, the Banach algebra p

  Then for any ✏ °0,o n eh a s ~a `b~s p " lim nÑ8 ~pa `bq n ~1 n " lim § max ~a~s p , ~b~s p , ✏ ( where the second inequality comes from the inequality of arithmetic and geometric means. Thus ~¨~s p is ultrametric. In view of Proposition 2.1.7,o n eo b t a i n st h a t~¨~s p is ultrametric and powermultiplicative. Proposition 2.1.13.-Let R ' "

				nÑ8	~n ÿ i"0 ˆn i	˙an´i b i ~1 n
	§ lim sup nÑ8 p max iPt0,...,nu	~an´i ~~b i ~(q	1 n
	§ lim sup nÑ8	max iPt0,...,nu	n	´i n	~an´i ~1 n´i	`i n	~bi ~1 i	(
	§ lim sup nÑ8	max iPt0,...,nu	n	´i n	C	1 n´i ✏	p~a~s p `✏q`i n	C	1 i ✏ p~b~s p `✏q (

  Proposition 2.1.16.-Let ~¨~be an ultrametric power-multiplicative graded seminorm on R ' " krT s. Then it identifies with the Gauss-Tate algebra norm of radius ~T ~, namely The spectrum Mp p R ' q is isomorphic to the closed disc of radius ~T ~in Spec krT s an .A point z P Spec krT s an belongs to this disc if and only if |T | z § ~T ~.

	@s "	ÿ nPN	n	¨T n , ~s~"max nPN	| n | ¨~T	~n( .
	Proof.-For any s P krT s, since the norm is ultrametric and power-multiplicative, one
	has
	~s~"max nPN	~sn ~( " max nPN	| n | ¨~T n ~( " max nPN	| n | ¨~T	~n( .
	The second assertion follows easily from this identification. For the third assertion, z
	belongs to the disc if and only if |¨| § C ¨~¨~.S o
	|T | z " lim nÑ8	|T n |	1 n § lim nÑ8	pC ¨~T n ~q 1 n "~T	~.
	Conversely if |T |

z § ~T ~, then |T | s § ~s~for any s P krT s thanks to the ultrametricity of |¨| z and the gradedness of ~¨~. Remark 2.1.17.-The above assertion is not true if one replaces krT s by krT 1 ,...,T d s for d • 2.

  bn pxq, from the scalar extension norm k¨k n,p pxq of k¨k n on R n pL qb k p pxq. The quotient norm is denoted by k¨k n,X|x .I f L bn is generated by global sections, the family k¨k n,X|x

~¨~ pxq is an ultrametric graded seminorms on Sym p pxq L pxq,o n eh a sw P D _ pL , q if and only if |e 1 | w § |e 1 | ,i fa n d only if w P D _ pL , qpxq by Proposition 2.1.16. 2.1.7. Fubini-Study metrics. -2.1.7.1. Fubini-Study metric associated with a norm. -Let L be an invertible O Xmodule. Let k¨k n be a norm on R n pL q.F o ra n yx P X an , if the evaluation map R n pL qb k p pxq ›Ñ L bn pxq is surjective, then it induces a quotient norm on the p pxq-vector space L

  ' pL qpxqÑSym p pxq L pxq is a (TN)-isomorphism, by Fact 2.1.18, one can find a unique graded powermultiplicative seminorm on Sym p pxq L pxq, denoted by ~¨~1 X|x;sp , whose restriction on R ' pL qpxq is ~¨~X |x;sp . Denote by p R ' pL , ~¨~qpxq the separated completion of pR ' pL qpxq, ~¨~X |x q. By construction, the quotient algebra norm ~¨~X |x is in fact the orthogonal sum algebra norm of the sub-multiplicative family tk¨k n,X|x u nPN . Proposition 2.1.23.-For any x P X an , the limit seminorm Pp~¨~qpxq of the family t 1 Proof.-Since the algebra seminorm ~¨~X |x is the orthogonal sum of the family k¨k n,X|x ( nPN , it is graded, so one has |e 1 | Pp~¨~q " lim

	nÑ8	|e 1 | 1 n FSpk¨knq " lim nÑ8	|e bn 1 pxq|	1 n FSpk¨knq
	" lim nÑ8	ke bn 1 pxqk	1 n n,X|x " lim nÑ8	~ebn 1 pxq~1 n X|x
	"~e N 1 pxq~1 N X|x;sp "~e 1 ~1 X|x;sp .

qbp pxq ›Ñ à nPN L bn pxq": R ' pL qpxq. The above quotient algebra homomorphism induces a quotient algebra norm on R ' pL qpxq, from the scalar extension norm ~¨~p pxq of ~¨~on R ' pL qbp pxq. The quotient algebra norm is denoted by ~¨~X |x . Its spectral seminorm, denoted by ~¨~X |x;sp , is power-multiplicative. As the inclusion of graded p pxq-algebras L : R n FSpk¨k n qpxqu nPN can be expressed by ~¨~X |x in the following way: @e 1 P L pxqzt0u, |e 1 | Pp~¨~q "~e 1 ~1 X|x;sp .

  FSpk¨knq " inf pµ 1 ,...,µ dn qPp pxq dn µ 1 s n,1 pxq`¨¨¨`µ dn s n,dn "en ,...,µ dn qPp pxq dn µ 1 s n,1 pxq`¨¨¨`µ dn s n,dn "en .2.2. Passages between norms and metrics. -Recall that we have operations passing between space of norms and space of metrics as follows tk¨k n on R n pL qu

	ÿ jPt1,...,dnu pµ 1 max inf jPt1,...,dnu inf max jPt1,...,dnu | j | x ¨| j | ´1 p µ j ¨sn,j |µ j | x ¨ks n,j k pxq ¨ks n,j k ( " min " " jPt1,...,dnu | j | ´1 x ¨ks n,j k n ( . The last equality is obtained with Lemma 2.2.2. Corollary 2.2.4.-With the same hypothesis as above, for e _ n Pp L bn q _ pxqzt0u,o n e has |e _ n | FSpk¨knq k¨k n o o

n | FSpk¨knq " min jPt1,...,dnu e n s n,j pxq x ¨ks n,j k n ( , with the convention that 0 ´1 "`8. (see also

[START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF] Lemma 3.3]

)

Proof.-For j Pt0,...,d n u, let j P p pxq be such that s n,j pxq" j ¨en . By construction,

|e n | p 1 ,..., dn qPp pxq dn  1 `¨¨¨` dn "1 max jPt1,...,dnu | j ¨ ´1 j | x ¨ks n,j k ( " inf p 1 ,..., dn qPp pxq dn  1 `¨¨¨` dn "1 _ " max |e _ n ps n,j q| x ¨ks n,j k ´1 n ( .

Proof.-It suffices to note that e _ n ps n,j q" j .

2

FSpk¨knq / / t n on L bn u

  Theorem 6.15 (iii)])Proof.-By assumption, there exists a family of norms tk¨k n u nPN such that uniformly for x P X an ,

	lim nÑ8 |p˚qpxq| and 1 n
	distp	1 n	FSpk¨k n q,	1 n	FSpk¨k

FSpk¨knq " |p˚qpxq| , so for any ✏ °0, there exists N 0 P N such that for any n • N 0 distpn , FSpk¨k n qq § n✏, hence by Lemma 2.2.5 and 2.2.6, distpFSpk¨k n q, FSpk¨k FSpk¨knq qq § n✏. By Proposition 2.2.9,o n eh a sFSpk¨k FSpk¨knq qq " FSpk¨k n q,s o distpFSpk¨k n q, FSpk¨k n qq § n✏, n qq § ✏.

  Proposition 2.3.1.-For any s n P R n pL q,o n eh a s Spectrum of normed section algebra in terms of envelope metric. -We relate the Berkovich spectrum of normed section algebra with the dual unit disc bundle with respect to the envelope metric.

	sup zPM ´p q	|s n | z "	sup wPD _ pL , q	|s n | w " sup xPX an	|s n pxq| n " ks n k n ,
	and for any s P R ' pL q,o n eh a s
		sup	|s| z " max
		zPM ´p q
						Thus
	the norm
		sup wPD _ pL , q |¨| nPN |¨| z " sup xPX an sup wPD _ pL , qpxq sup wPD _ pL , q |s n | w ( " max nPN ks n k n	(	"~s~ .
	2.3.2. Proposition 2.3.2.

nPN

ks n k n "~s~ .

Proof.-The first equality follows immediately from the construction. For the second assertion, remember that we denote by D _ pL , qpxq the fiber over x of

p an : D _ pL ,

qÑX an , it is contained in Spec p pxq Sym p pxq L pxq an as a disc. Note that for any s P S ' pL q,o n eh a s sup wPD _ pL , q |s| w " sup xPX an sup wPD _ pL , qpxq |s| w . Since D _ pL , qpxq is a disc over p pxq, by Corollary 2.1.16,t h en o r m sup wPD _ pL , qpxq |¨| w

is graded on the graded p pxq-algebra Sym p pxq L pxq, hence is a graded norm when restricted to the p pxq-algebra R ' pL qpxq and is also graded on the graded k-algebra R ' pL q. w is also graded on the graded k-algebra R ' pL q. Thanks to the first assertion, this gradedness of the norm implies that sup wPD _ pL , q |s| w " max

  1 n FSpk¨k n qq (resp. to D _ pL , 1 n FSpk¨k n qq) if and only if for any point x P X an , and any local section e 1 of L,o n eh a s |e 1 | w § |e 1 | 1 n FSpk¨knq . or equivalently, for any local section e n of L bn ,o n eh a s |e n | w § |e n | FSpk¨knq . If this holds, then by Proposition 2.2.3, for any j Pt 0,...,d n u, one considers the local sections given by s n,j ,a n do n eh a s |s n,j pxq| w § |s n,j pxq| FSpk¨knq " min iPt1,...,dnu s n,j pxq s n,i pxq x ¨ks n,i k n ( § ks n,j k n . Conversely, if these inequality holds, then for any j Pt0,...,d n u, and any local section e n such that e n ‰ 0,o n eh a s

	|e n | w " |	e n s n,j pxq	| x ¨|s n,j pxq| w § |	e n s n,j pxq	| p pxq ¨ks n,j k n
	which implies, by Proposition 2.2.3 that
	|e n | w § min jPt1,...,dnu	e n s n,j pxq x ¨ks n,j k n	(	" |e n | FSpk¨knq .

  , the second norm being the spectral algebra norm of the first by Corollary 2.4.4. We've fixed M P N such that for any n • M , L bn is very ample, and that the restriction maps are surjective.The starting point is the spectral relation between these two algebra norms as shown in Corollary 2.4.4, ~¨~ ,X|Y ;sp " ~¨~ | Y , which is a consequence of the geometric homeomorphism of spectra Mp p R ' pL X|Y , X|Y qq » Mp p R ' pL X|Y , | Y qq. Theorem 3.2.1.-Let be a semipositive metric on L , then for any ✏ °0,andanyt 1 P H 0 pY, L | Y q, there exists n Y P N such that for any n • n Y , there exists s n P H 0 pX, L bn q with s n | Y " t bn 1 and ks n k n § e n✏ ¨pkt 1 k | Y q n . comparison stems from holomorphic functional calculus, which produces a homomorphism of Banach algebras W Ñ p R ' pL X|Y , X|Y q. The comparison of algebra norms follows from the continuity of this homomorphism. The second comparison is easy: since both algebra norms ~¨~W ✏ and ~¨~ | Y p✏q are sup norms on W ✏ and on Mp p R ' pL X|Y , | Y p✏qqq respectively, the desired comparison follows directly from the inclusion of spectra. 3.3.1. Localization of spectrum by affinoid domain covering. -For any ✏ °0, one considers the spectra Mp p R ' pL X|Y , X|Y qq and Mp p R ' pL X|Y , | Y p✏qqq. They can be both indentified with closed subsets in CpL X|Y q an , and one denotes by Int t C the topological interior of a set in CpL X|Y q an . Lemma 3.3.1.-There is an affinoid algebra norm ~¨~3 ✏p✏q,X|Y on R ' pL X|Y q such that ~¨~3 ✏ p✏q,X|Y • ~¨~ p✏q,X|Y . Denote by p R ' pL X|Y , 3 ✏ q the Banach algebra completion, it is an affinoid algebra, and the identity map on R ' pL X|Y q induces a continuous homomorphismp R ' pL X|Y , 3 ✏ qÑ p R ' pL X|Y , X|Y p✏qq.which induces an inclusion of spectraMp p R ' pL X|Y , X|Y p✏qqq Ñ Mp p R ' pL X|Y , 3 ✏ qq.Proof.-The existence of affinoid norm is an immediate consequence of Proposition 1.3.7 applied to the Banach algebra p R ' pL X|Y , X|Y p✏q and its finitely generated subalgebra R ' pL X|Y q. The induced map on spectra is injective by Proposition 1.2.11.Recall that a special domain is a finite union of affinoid domains (see Definition 1.3.36). 3.3.2.-Assume that | Y is semipositive (in particular, it is continuous). For any ✏ °0, there exists a special domain W ✏ such thatMp p R ' pL X|Y , X|Y qq » Mp p R ' pL X|Y , | Y qq Ñ W ✏ Ñ Mp p R ' pL X|Y , | Y p✏qqq.Proof.-The first isomorphism is given by Proposition 2.4.3. To construct W ✏ , note that by Corollary 1.3.29, every point in CpR ' pL X|Y , p✏q X|Y qq an has a neighbourhood system consisting of affinoid domains. By Proposition 2.3.9, the open set Int t C pMp p R ' pL X|Y , | Y p✏qqqq is a neighbourhood of Mp p R ' pL X|Y , | Y qq, because the metric | Y is continuous. Thus for any z P Mp p R ' pL X|Y , X|Y qq,onema yfindanaffinoid domain neighbourhood V pzq which is contained in Int t C pMp p R ' pL X|Y , | Y p✏qqqq. Since the left hand side is a compact set by Proposition 1.2.6, there exist finitely many points tz 1 ,...,z m u such that tInt t C V pz i qu iPt1,...,mu form a covering of Mp p R ' pL X|Y , X|Y qq. Let W ✏ be the union of affinoid domains tV pz i qu iPt1,...,mu , then it is a special domain, and satisfies the desired inclusion conditions. Remark 3.3.3.-One denotes by W ✏ the structural Banach k-algebra ΓpW ✏ , O W✏ q of the special domain W ✏ , equipped with supremum norm ~¨~W ✏ (see Definition 1.3.36).

	Proposition (see the picture
	below)
	One forms a covering by open sets Mp p R ' pL X|Y , X|Y qq Ñ	§	Int t C V pzq.
		zPMp p R'pL X|Y , X|Y qq

  r rqq " inf p 0 ¨T0 `¨¨¨` d ¨Td qpxpr r rqq"pT i qpxpr r rqq p 0 ,..., d qPk d`1

				0 ¨T0 `¨¨¨` d ¨Td
				!	)
	"	inf p 0 ,..., d qPk d`1 0 ¨0 `¨¨¨` d ¨d " i	max mPt0,...,du	k m ¨Tm k
	"	inf p 0 ,..., d qPk d`1 0 ¨0 `¨¨¨` d ¨d " i	max mPt0,...,du

  Proposition 3.4.6.-Assume that is a Fubini-Study metric. The conclusion of Proposition 3.4.3 holds without the assumption of pk, |¨|q-freeness of kT i k 1 ( iPt0,...,du . Proof.-Since |¨| k is discrete, for any ✏ °0, there exist with | | § ✏ such that the elements kT i k p q ( iPt0,...,du are pk, |¨|q-free. By Proposition 3.4.5, for any n P N and any s n P R n pOp1qq which one writes as ∞ |J|"n J ¨T T T J ,o n eh a s

	e ´n✏	ÿ
		|J|"n

  3.4.1.2. Case for some general pX, L q.-Proposition 3.4.7.-Assume that is a (first level) Fubini-Study metric. If L is very ample, then p R ' pL , q, p R ' pL X|Y , X|Y q and p R ' pL | Y , | Y q are affinoid algebras.

  Remark 3.4.9.-With the metric finiteness properties of affinoid algebra norm, here the upper bound for metric extension of a Fubini-Study metric is much better than the bound (4) or even the bound (3), for its (in)depence on n P N. 3.4.2. Algebra norm induced by semipositive metric. -With the extra assumption of discreteness of valuation of the base valued field, one would be able to give another proof of Theorem 3.3.5,i fP r o p o s i t i o n3.4.7 holds for ample L and any m-th Fubini-Study metric . Recall that we've fixed M P N such that the restriction maps R n pL qÑR n pL | Y q are surjective for all n • M . Theorem 3.4.10.-Suppose that pk, |¨|q is discretely valued. Assume that Proposition 3.4.7 holds for ample L and any m-th Fubini-Study metric . Let be a semipositive metric on L . Then for any ✏ °0, there exist n Y P N such that for any n • n Y and any t n P H 0 pY, L | bn Y q, there exits s n P H 0 pX, L bn q such that s n | Y " t n and ks n k n § e n✏ ¨kt n k n | Y .

		2.1.26,b y
	Proposition 1.3.22, there exist Cp ,Yq °0 such that
	(5)	~¨~

X|Y § Cp ,Y,Xq ¨~¨~ X|Y ;sp . Since ~¨~ X|Y ;sp " ~¨~ | Y by Corollary 2.4.4, one gets the desired estimate.

Remerciements

Since the k-linear map R n pL qÑR n pL X|Y q is surjective for all large n P N,andk¨k n ,X|Y is the quotient norm of k¨k n ,o n eh a s

FSpk¨k n qpyq"FSpk¨k n ,X|Y qpyq.

Hence the two envelope metrics are equal. Proof.-Suppose that is the pointwise limit on X an of t 1 n FSpk¨k n qu, where tk¨k n u are norms on R n pL q. Then L | Y is the pointwise limit of t

Note that ~¨~ ,X|Y is submultiplicative and ~¨~ | Y is power-multiplicative, so the sequence

,X|Y u n•M has a limit and one deduces that

Hence for any ✏ °0 there exists n Y P N, such that for any n • n Y , there exists

-This result is previously obtained in [START_REF] Chen | Extension property of semipositive invertible sheaves over a non-archimedean field[END_REF], by using approximation of by model metrics. Here we give another proof.

Geometric/Spectral approximation

Recall that CpL X|Y q is the affine scheme Spec R ' pL X|Y q. The spectra of p R ' pL X|Y , X|Y q and p R ' pL X|Y , | Y q are compact sets in CpL X|Y q an . This approach deduces comparison of algebra norms on R ' pL X|Y q from the geometric comparison of the spectra of the corresponding Banach algebras that are completions with respect to these norms. In the previous section, we've seen that a homeomorphism of spectra

gives rise to a weak version of comparison. To refine it to a uniform one, we examine the strict inclusion of spectra

and the consequent comparison of algebra norms. Here ~¨~ | Y p✏q is the sup algebra norm associated with the dilated metric p✏q.

The strategy is as follows: starting from the above inclusion of spectra, we can find a special domain W ✏ contained between them

which we see as a geometric approximation of Mp p R ' pL X|Y , X|Y qq. We consider an intermediate Banach algebra W, which is the structural Banach algebra of W ✏ equipped with the sup algebra norm ~¨~W ✏ . We would like to translate the above inclusions of spectra to a comparison of corresponding algebra norms of the following form:

where we use the symbol À to mean § which omits constants that depends only on the background geometric data (X, Y , L and ). The first comparison is non-trivial, since it bounds a general quotient norm ~¨~ ,X|Y from above by a sup norm ~¨~W ✏ . This Proof.-Without loss of generality, one may assume that L is very ample. One takes the affine analytic space CpL q an as Z.F o ra n y✏ P R, consider the compact set M ´p p✏qq. Then by Proposition 2.3.4,o n eh a s { M ´p p✏qq " Mp p✏qq.

Note that for ✏ °0, by the continuity of , M ´p p✏qq is a neighbourhood of M ´p q in V . Hence by Speculation 4.2.1, the set Mp p✏qq is a neighbourhood of Mp q in Z. By Corollary 4.3.1, the envelope metric Pp q is continuous.