logo AFST
Approches courantielles à la Mellin dans un cadre non archimédien
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 2, pp. 357-396.

On propose une approche du type Mellin pour l’approximation des courants d’intégration ou la réalisation effective de courants de Green normalisés associés à un cycle 1 m [div(s j )], où s j est une section méromorphe d’un fibré en droites j U au-dessus d’un ouvert U d’un bon espace de Berkovich, lorsque chaque j est équipé d’une métrique lisse et que codim U ( jJ Supp[div(s j )])#J pour tout ensemble J{1,,p}. On étudie aussi la transposition au cadre non archimédien des formules de Crofton et de King, en particulier la réalisation approchée de courants de Vogel et de Segre.

We propose an approach of Mellin type for the approximation of integration currents or the effective realization of normalized Green currents associated with a cycle 1 m [div(s j )], where s j is a meromorphic section of a line bundle j U over an open U in a good Berkovich space when each j has a smooth metric and codim U ( jJ Supp[div(s j )])#J for every set J{1,,p}. We also study the transposition to the non archimedean context of Crofton and King formulas, particularly the approximate realization of Vogel and Segre currents.

Reçu le : 2016-04-07
Accepté le : 2017-05-22
Publié le : 2019-05-02
DOI : https://doi.org/10.5802/afst.1602
Classification : 32U25,  32U35,  32U40,  14G22,  14G40,  14TXX
Mots clés: courants, diviseurs, équations de Lelong–Poincaré, formule de King, nombres de Lelong
@article{AFST_2019_6_28_2_357_0,
     author = {Ibrahima Hamidine},
     title = {Approches courantielles \`a la Mellin dans un cadre non archim\'edien},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {6e s{\'e}rie, 28},
     number = {2},
     year = {2019},
     pages = {357-396},
     doi = {10.5802/afst.1602},
     zbl = {07095685},
     mrnumber = {3957684},
     language = {fr},
     url = {afst.centre-mersenne.org/item/AFST_2019_6_28_2_357_0/}
}
Ibrahima Hamidine. Approches courantielles à la Mellin dans un cadre non archimédien. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 2, pp. 357-396. doi : 10.5802/afst.1602. https://afst.centre-mersenne.org/item/AFST_2019_6_28_2_357_0/

[1] Mats Andersson; Håkan Samuelsson Kalm; Elizabeth Wulcan; Alain Yger Segre numbers, a generalized King formula, and local intersections, J. Reine Angew. Math., Volume 728 (2017), pp. 105-136 | MR 3668992 | Zbl 1375.32020

[2] Farhad Babaee Complex Tropical Currents : Extremality, and Approximation (https://arxiv.org/abs/1403.7456)

[3] Farhad Babaee; June Huh A tropical approach to a generalized Hodge conjecture for positive currents, Duke Math. J., Volume 166 (2017) no. 14, pp. 2749-2813 | MR 3707289 | Zbl 1396.14064

[4] Carlos A. Berenstein; Roger Gay; Alekos Vidras; Alain Yger Residue currents and Bezout identities, Progress in Mathematics, Volume 114, Birkhäuser, 1993 | MR 1249478 | Zbl 0802.32001

[5] Carlos A. Berenstein; Alain Yger Green currents and analytic continuation, J. Anal. Math., Volume 75 (1998), pp. 1-50 | MR 1655822 | Zbl 0910.14009

[6] Vladimir G. Berkovich Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, Volume 33, American Mathematical Society, 1990 | MR 1070709 | Zbl 0715.14013

[7] Sébastien Boucksom; Charles Favre; Mattias Jonsson Singular semipositive metrics in non-Archimedean geometry, J. Algebr. Geom., Volume 25 (2016) no. 1, pp. 77-139 | MR 3419957 | Zbl 1346.14065

[8] José Ignacio Burgos Gil; Patrice Philippon; Martín Sombra Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, Volume 360, Société Mathématique de France, 2014 | Zbl 1311.14050

[9] Antoine Chambert-Loir Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl 1112.14022

[10] Antoine Chambert-Loir Heights and measures on analytic spaces. A survey of recent results, and some remarks, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II (London Mathematical Society Lecture Note Series) Volume 384, Cambridge University Press, 2011, pp. 1-50 | MR 2885340 | Zbl 1279.14027

[11] Antoine Chambert-Loir Differential forms and currents on Berkovich spaces, 2013 (lecture at the Simons Symposium on Nonarchimedean and tropical geometry, held in St John)

[12] Antoine Chambert-Loir; Antoine Ducros Formes différentielles réelles et courants sur les espaces de Berkovich (https://arxiv.org/abs/1204.6277v1)

[13] Brian Conrad Irreducible components of rigid spaces, Ann. Inst. Fourier, Volume 49 (1999) no. 2, pp. 473-541 | MR 1697371 | Zbl 0928.32011

[14] Antoine Ducros Variation de la dimension relative en géométrie analytique p-adique, Compos. Math., Volume 143 (2007) no. 6, pp. 1511-1532 | MR 2371379 | Zbl 1161.14018

[15] Terence Gaffney; Robert Gassler Segre numbers and hypersurface singularities, J. Algebr. Geom., Volume 8 (1999) no. 4, pp. 695-736 | MR 1703611 | Zbl 0971.13021

[16] Walter Gubler Equidistribution over function fields, Manuscr. Math., Volume 127 (2008) no. 4, pp. 485-510 | MR 2457191 | Zbl 1189.14030

[17] Walter Gubler Forms and current on the analytification of an algebraic variety (after Chambert-Loir and Ducros), Nonarchimedean and tropical geometry (Simons Symposia), Springer, 2016, pp. 1-30 | MR 3700066 | Zbl 1349.14100

[18] Walter Gubler; Klaus Künnemann A tropical approach to nonarchimedean Arakelov geometry, Algebra Number Theory, Volume 11 (2017) no. 1, pp. 77-180 | MR 3602767 | Zbl 1386.14096

[19] Jun-ichi Igusa An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, Volume 14, American Mathematical Society ; International Press, 2000 | MR 1743467 | Zbl 0959.11047

[20] Xinyi Yuan Algebraic dynamics, canonical heights and Arakelov geometry, Fifth International Congress of Chinese Mathematicians. Part 2 (AMS/IP Studies in Advanced Mathematics) Volume 51-2, American Mathematical Society, 2012, pp. 893-929 | MR 2918034 | Zbl 1247.14026