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 , where is a meromorphic section of a line bundle over an open in a good Berkovich space when each has a smooth metric and for every set . We also study the transposition to the non archimedean context of Crofton and King formulas, particularly the approximate realization of Vogel and Segre currents.
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 , où est une section méromorphe d’un fibré en droites au-dessus d’un ouvert d’un bon espace de Berkovich, lorsque chaque est équipé d’une métrique lisse et que pour tout ensemble . 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.
Accepted:
Published online:
DOI: 10.5802/afst.1602
Mot clés : courants, diviseurs, équations de Lelong–Poincaré, formule de King, nombres de Lelong
Keywords: currents, divisors, Lelong–Poincaré equations, King formula, Lelong numbers
Ibrahima Hamidine 1
@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}, pages = {357--396}, publisher = {Universit\'e Paul Sabatier, Toulouse}, volume = {6e s{\'e}rie, 28}, number = {2}, year = {2019}, doi = {10.5802/afst.1602}, mrnumber = {3957684}, zbl = {07095685}, language = {fr}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1602/} }
TY - JOUR AU - Ibrahima Hamidine TI - Approches courantielles à la Mellin dans un cadre non archimédien JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2019 SP - 357 EP - 396 VL - 28 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1602/ DO - 10.5802/afst.1602 LA - fr ID - AFST_2019_6_28_2_357_0 ER -
%0 Journal Article %A Ibrahima Hamidine %T Approches courantielles à la Mellin dans un cadre non archimédien %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2019 %P 357-396 %V 28 %N 2 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1602/ %R 10.5802/afst.1602 %G fr %F 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, Serie 6, Volume 28 (2019) no. 2, pp. 357-396. doi : 10.5802/afst.1602. https://afst.centre-mersenne.org/articles/10.5802/afst.1602/
[1] Segre numbers, a generalized King formula, and local intersections, J. Reine Angew. Math., Volume 728 (2017), pp. 105-136 | MR | Zbl
[2] Complex Tropical Currents : Extremality, and Approximation (https://arxiv.org/abs/1403.7456) | Zbl
[3] A tropical approach to a generalized Hodge conjecture for positive currents, Duke Math. J., Volume 166 (2017) no. 14, pp. 2749-2813 | DOI | MR | Zbl
[4] Residue currents and Bezout identities, Progress in Mathematics, 114, Birkhäuser, 1993 | MR | Zbl
[5] Green currents and analytic continuation, J. Anal. Math., Volume 75 (1998), pp. 1-50 | DOI | MR | Zbl
[6] Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990 | MR | Zbl
[7] Singular semipositive metrics in non-Archimedean geometry, J. Algebr. Geom., Volume 25 (2016) no. 1, pp. 77-139 | MR | Zbl
[8] Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, 360, Société Mathématique de France, 2014 | Zbl
[9] Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Volume 595 (2006), pp. 215-235 | Zbl
[10] 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 | Zbl
[11] Differential forms and currents on Berkovich spaces, 2013 (lecture at the Simons Symposium on Nonarchimedean and tropical geometry, held in St John) | Zbl
[12] Formes différentielles réelles et courants sur les espaces de Berkovich (https://arxiv.org/abs/1204.6277v1) | Zbl
[13] Irreducible components of rigid spaces, Ann. Inst. Fourier, Volume 49 (1999) no. 2, pp. 473-541 | DOI | MR | Zbl
[14] Variation de la dimension relative en géométrie analytique -adique, Compos. Math., Volume 143 (2007) no. 6, pp. 1511-1532 | DOI | MR | Zbl
[15] Segre numbers and hypersurface singularities, J. Algebr. Geom., Volume 8 (1999) no. 4, pp. 695-736 | MR | Zbl
[16] Equidistribution over function fields, Manuscr. Math., Volume 127 (2008) no. 4, pp. 485-510 | DOI | MR | Zbl
[17] 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 | Zbl
[18] A tropical approach to nonarchimedean Arakelov geometry, Algebra Number Theory, Volume 11 (2017) no. 1, pp. 77-180 | MR | Zbl
[19] An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, 14, American Mathematical Society ; International Press, 2000 | MR | Zbl
[20] 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 | Zbl
Cited by Sources: