Metrised ample line bundles in non-Archimedean geometry
Annales de l'Institut Fourier, Online first, 37 p.

We study metrised ample line bundles on projective varieties over non-Archimedean fields from the point of view of commutative Banach algebras and analytic functions of several variables. Line bundle metrics induce sup norms on the graded algebra of sections; the global metric positivity is interpreted as the holomorphic convexity of the spectrum of the normed section algebra. A normed extension property is established using spectral and functional methods: restricted sections on a closed subvariety can be extended to sections on the ambient variety, with a sub-exponential asymptotic distorsion of sup norms.

On étudie fibrés en droites amples métrisés sur variétés projectives définies sur un corps non-archimédien, d’un point de vue d’algèbres de Banach commutatives et fonctions analytiques à plusieurs variables. Une métrique sur un fibré en droites induit une norme sup en l’algèbre graduée de sections ; la positivité au sens global de cette métrique est interprétée comme la convexité holomorphe du spectre analytique de cette algèbre normée de sections. Une propriété d’extension normée est établie par techniques spectrale et fonctionnel : on peut étendre une section restreinte sur une sous-variété en une section sur la variété ambiante, avec un contrôle sous-exponentiel de la distorsion asymptotique des normes sup.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3549
Classification: 32P05,  32E20,  46H30
Keywords: Non-Archimedean geometry, holomorphic convexity, functional calculus.
Fang, Yanbo 1

1 IMJ-PRG Université de Paris Paris, 75013 (France)
@unpublished{AIF_0__0_0_A130_0,
     author = {Fang, Yanbo},
     title = {Metrised ample line bundles in {non-Archimedean} geometry},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2022},
     doi = {10.5802/aif.3549},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Fang, Yanbo
TI  - Metrised ample line bundles in non-Archimedean geometry
JO  - Annales de l'Institut Fourier
PY  - 2022
DA  - 2022///
PB  - Association des Annales de l’institut Fourier
N1  - Online first
UR  - https://doi.org/10.5802/aif.3549
DO  - 10.5802/aif.3549
LA  - en
ID  - AIF_0__0_0_A130_0
ER  - 
%0 Unpublished Work
%A Fang, Yanbo
%T Metrised ample line bundles in non-Archimedean geometry
%J Annales de l'Institut Fourier
%D 2022
%I Association des Annales de l’institut Fourier
%Z Online first
%U https://doi.org/10.5802/aif.3549
%R 10.5802/aif.3549
%G en
%F AIF_0__0_0_A130_0
Fang, Yanbo. Metrised ample line bundles in non-Archimedean geometry. Annales de l'Institut Fourier, Online first, 37 p.

[1] Abbes, A.; Bouche, T. Théorème de Hilbert–Samuel «  arithmétique  » , Ann. Inst. Fourier (Grenoble), Volume 45 (1995) no. 2, pp. 375-401 | DOI | MR | Zbl

[2] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990, x+169 pages | DOI | MR | Zbl

[3] Bloch, S.; Gillet, H.; Soulé, C. Non-Archimedean Arakelov theory, J. Algebraic Geom., Volume 4 (1995) no. 3, pp. 427-485 | MR | Zbl

[4] Bosch, S.; Güntzer, U.; Remmert, R. Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 261, Springer-Verlag, Berlin, 1984, xii+436 pages | DOI | MR | Zbl

[5] Bost, J.-B. Germs of analytic varieties in algebraic varieties: canonical metrics and arithmetic algebraization theorems, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 371-418 | MR | Zbl

[6] Bouche, Thierry Convergence de la métrique de Fubini–Study d’un fibré linéaire positif, Ann. Inst. Fourier (Grenoble), Volume 40 (1990) no. 1, pp. 117-130 | DOI | MR | Zbl

[7] Boucksom, Sébastien; Eriksson, Dennis Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, Adv. Math., Volume 378 (2021), 107501, 124 pages | DOI | MR | Zbl

[8] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom., Volume 25 (2016) no. 1, pp. 77-139 | DOI | MR | Zbl

[9] Bourbaki, Nicolas Espaces vectoriels topologiques. Chapitres 1 à 5, Éléments de mathématique. [Elements of mathematics], Masson, Paris, 1981, p. vii+368 | MR | Zbl

[10] Burgos Gil, José Ignacio; Gubler, Walter; Jell, Philipp; Künnemann, Klaus; Martin, Florent Differentiability of non-archimedean volumes and non-archimedean Monge–Ampère equations (With an appendix by Robert Lazarsfeld), Algebr. Geom., Volume 7 (2020) no. 2, pp. 113-152 | DOI | MR | Zbl

[11] Burgos Gil, José Ignacio; Moriwaki, Atsushi; Philippon, Patrice; Sombra, Martín Arithmetic positivity on toric varieties, J. Algebraic Geom., Volume 25 (2016) no. 2, pp. 201-272 | DOI | MR | Zbl

[12] Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, Société Mathématique de France, Paris, 2014 no. 360, vi+222 pages | MR | Zbl

[13] Chambert-Loir, Antoine Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | DOI | MR | Zbl

[14] Chambert-Loir, Antoine; Ducros, A. Formes différentielles réelles et courants sur les espaces de Berkovich (2012) (https://arxiv.org/abs/1204.6277)

[15] Chen, Huayi; Moriwaki, Atsushi Extension property of semipositive invertible sheaves over a non-archimedean field, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 18 (2018) no. 1, pp. 241-282 | DOI | MR | Zbl

[16] Chen, Huayi; Moriwaki, Atsushi Arakelov geometry over adelic curves, Lecture Notes in Mathematics, 2258, Springer, Singapore, [2020] ©2020, xviii+450 pages | DOI | MR | Zbl

[17] Demailly, J.-P. Complex analytic and differential geometry, Institut Fourier, 2012

[18] Grauert, Hans Charakterisierung der holomorph vollständigen komplexen Räume, Math. Ann., Volume 129 (1955), pp. 233-259 | DOI | MR | Zbl

[19] Grothendieck, A. Éléments de géométrie algébrique II, Inst. Hautes Études Sci. Publ. Math. (1960–1967) no. 4,8,11,17,20,24,28,32

[20] Gubler, Walter Local heights of subvarieties over non-Archimedean fields, J. Reine Angew. Math., Volume 498 (1998), pp. 61-113 | DOI | MR | Zbl

[21] Gubler, Walter; Künnemann, Klaus Positivity properties of metrics and delta-forms, J. Reine Angew. Math., Volume 752 (2019), pp. 141-177 | DOI | MR | Zbl

[22] Gubler, Walter; Martin, Florent On Zhang’s semipositive metrics, Doc. Math., Volume 24 (2019), pp. 331-372 | DOI | MR | Zbl

[23] Maculan, Marco; Poineau, Jérôme Notions of Stein spaces in non-Archimedean geometry, J. Algebraic Geom., Volume 30 (2021) no. 2, pp. 287-330 | DOI | MR | Zbl

[24] Manivel, Laurent Un théorème de prolongement L 2 de sections holomorphes d’un fibré hermitien, Math. Z., Volume 212 (1993) no. 1, pp. 107-122 | DOI | MR | Zbl

[25] Moriwaki, Atsushi Free basis consisting of strictly small sections, Int. Math. Res. Not. IMRN (2011) no. 6, pp. 1245-1267 | DOI | MR | Zbl

[26] Randriambololona, Hugues Métriques de sous-quotient et théorème de Hilbert–Samuel arithmétique pour les faisceaux cohérents, J. Reine Angew. Math., Volume 590 (2006), pp. 67-88 | DOI | MR | Zbl

[27] Serre, Jean-Pierre Faisceaux algébriques cohérents, Ann. of Math. (2), Volume 61 (1955), pp. 197-278 | DOI | MR | Zbl

[28] Tian, Gang On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., Volume 32 (1990) no. 1, pp. 99-130 http://projecteuclid.org/euclid.jdg/1214445039 | DOI | MR | Zbl

[29] Vishik, M. M. Non-Archimedean spectral theory, Current problems in mathematics (Itogi Nauki i Tekhniki), Volume 25, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 51-114 | MR

[30] Zhang, Shouwu Positive line bundles on arithmetic varieties, J. Amer. Math. Soc., Volume 8 (1995) no. 1, pp. 187-221 | DOI | MR | Zbl

[31] Zhang, Shouwu Small points and adelic metrics, J. Algebraic Geom., Volume 4 (1995) no. 2, pp. 281-300 | MR | Zbl

Cited by Sources: