Seshadri constants and interpolation on commutative algebraic groups
Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1269-1289.

In this article we study interpolation estimates on a special class of compactifications of commutative algebraic groups constructed by Serre. We obtain a large quantitative improvement over previous results due to Masser and the first author and our main result has the same level of accuracy as the best known multiplicity estimates. The improvements come both from using special properties of the compactifications which we consider and from a different approach based upon Seshadri constants and vanishing theorems.

Dans cet article on étudie les lemmes d’interpolation dans les compactifications à la Serre de groupes algébriques commutatifs. On obtient un résultat aussi précis que les meilleurs lemmes de multiplicité connus, ce qui améliore notablement le lemme d’interpolation de Masser et celui du premier auteur. Ce raffinement provient d’une approche différente, fondée sur les constantes de Seshadri et les théorèmes d’annulation, et utilise les propriétés particulières des compactifications considérées.

DOI: 10.5802/aif.2880
Classification: 14L10, 14C20, 11J95, 14L40
Keywords: Interpolation estimate, Seshadri constant, ample line bundle, commutative algebraic group, obstruction subgroup, Seshadri exceptional subvariety
Mot clés : lemme d’interpolation, constante de Seshadri, fibré ample, groupe algébrique commutatif, sous-groupe obstructeur, sous-variété exceptionnelle de Seshadri

Fischler, Stéphane 1; Nakamaye, Michael 2

1 Univ Paris-Sud Laboratoire de Mathématiques d’Orsay CNRS, F-91405 Orsay (France)
2 Department of Mathematics and Statistics University of New Mexico Albuquerque, New Mexico 87131 (U.S.A.)
@article{AIF_2014__64_3_1269_0,
     author = {Fischler, St\'ephane and Nakamaye, Michael},
     title = {Seshadri constants and interpolation on commutative algebraic groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1269--1289},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {3},
     year = {2014},
     doi = {10.5802/aif.2880},
     zbl = {1330.14076},
     mrnumber = {3330170},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2880/}
}
TY  - JOUR
AU  - Fischler, Stéphane
AU  - Nakamaye, Michael
TI  - Seshadri constants and interpolation on commutative algebraic groups
JO  - Annales de l'Institut Fourier
PY  - 2014
SP  - 1269
EP  - 1289
VL  - 64
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2880/
DO  - 10.5802/aif.2880
LA  - en
ID  - AIF_2014__64_3_1269_0
ER  - 
%0 Journal Article
%A Fischler, Stéphane
%A Nakamaye, Michael
%T Seshadri constants and interpolation on commutative algebraic groups
%J Annales de l'Institut Fourier
%D 2014
%P 1269-1289
%V 64
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2880/
%R 10.5802/aif.2880
%G en
%F AIF_2014__64_3_1269_0
Fischler, Stéphane; Nakamaye, Michael. Seshadri constants and interpolation on commutative algebraic groups. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1269-1289. doi : 10.5802/aif.2880. https://aif.centre-mersenne.org/articles/10.5802/aif.2880/

[1] Birkenhake, Christina; Lange, Herbert Complex abelian varieties, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 302, Springer-Verlag, Berlin, 2004, pp. xii+635 | MR | Zbl

[2] Campana, Frédéric; Peternell, Thomas Algebraicity of the ample cone of projective varieties, J. Reine Angew. Math., Volume 407 (1990), pp. 160-166 | EuDML | MR | Zbl

[3] Ein, Lawrence; Küchle, Oliver; Lazarsfeld, Robert Local positivity of ample line bundles, J. Differential Geom., Volume 42 (1995) no. 2, pp. 193-219 | MR | Zbl

[4] Fischler, S. Interpolation on algebraic groups, Compos. Math., Volume 141 (2005) no. 4, pp. 907-925 | MR | Zbl

[5] Fischler, S.; Nakamaye, M. Connecting interpolation and multiplicity estimates in commutative algebraic groups (preprint arxiv 1209.2354 [math.NT], submitted) | Zbl

[6] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 2, Springer-Verlag, Berlin, 1998, pp. xiv+470 | MR | Zbl

[7] Hartshorne, Robin Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, pp. xvi+496 (Graduate Texts in Mathematics, No. 52) | MR | Zbl

[8] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 283-360 | MR | Zbl

[9] Knop, F.; Lange, H. Some remarks on compactifications of commutative algebraic groups, Comment. Math. Helv., Volume 60 (1985) no. 4, pp. 497-507 | EuDML | MR | Zbl

[10] Lazarsfeld, Robert Positivity in algebraic geometry. I and II, Ergebnisse der Mathematik und ihrer Grenzgebiete, 48, 49, Springer-Verlag, Berlin, 2004, pp. xviii+387, xviii+385 | MR | Zbl

[11] Masser, D. W. Interpolation on group varieties, Approximations diophantiennes et nombres transcendants (Luminy, 1982) (Progr. Math.), Volume 31, Birkhäuser, Boston, Mass., 1983, pp. 151-171 | MR | Zbl

[12] Masser, D. W.; Wüstholz, G. Zero estimates on group varieties. I, Invent. Math., Volume 64 (1981) no. 3, pp. 489-516 | MR | Zbl

[13] Mumford, David Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970, pp. viii+242 | MR | Zbl

[14] Nakamaye, Michael Multiplicity estimates and the product theorem, Bull. Soc. Math. France, Volume 123 (1995) no. 2, pp. 155-188 | Numdam | MR | Zbl

[15] Nakamaye, Michael Seshadri constants at very general points, Trans. Amer. Math. Soc., Volume 357 (2005) no. 8, pp. 3285-3297 | MR | Zbl

[16] Nakamaye, Michael Multiplicity estimates on commutative algebraic groups, J. Reine Angew. Math., Volume 607 (2007), pp. 217-235 | MR | Zbl

[17] Nakamaye, Michael Multiplicity estimates, interpolation, and transcendence theory, Number theory, analysis and geometry: In Memory of Serge Lang, D. Goldfeld et al. (ed), Springer, New York, 2012, pp. 475-498 | MR | Zbl

[18] Nakamaye, Michael; Ratazzi, Nicolas Lemmes de multiplicités et constante de Seshadri, Math. Z., Volume 259 (2008) no. 4, pp. 915-933 | MR | Zbl

[19] Philippon, Patrice Lemmes de zéros dans les groupes algébriques commutatifs, Bull. Soc. Math. France, Volume 114 (1986) no. 3, pp. 355-383 | Numdam | MR | Zbl

[20] Serre, J.-P. Quelques propriétés des groupes algébriques commutatifs, Astérisque (1978) no. 69–70, pp. 191-202

[21] Waldschmidt, M. Dépendance de logarithmes dans les groupes algébriques, Approximations diophantiennes et nombres transcendants (Luminy, 1982) (Progr. Math.), Volume 31, Birkhäuser, Boston, Mass., 1983, pp. 289-328 | MR | Zbl

[22] Waldschmidt, Michel La transformation de Fourier-Borel : une dualité en transcendance (lecture given in Delphes, September 1989, available from http://www.math.jussieu.fr/~miw)

[23] Waldschmidt, Michel Fonctions auxiliaires et fonctionnelles analytiques. I, II, J. Analyse Math., Volume 56 (1991), p. 231-254, 255–279 | MR | Zbl

[24] Waldschmidt, Michel Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 326, Springer-Verlag, Berlin, 2000, pp. xxiv+633 (Transcendence properties of the exponential function in several variables) | MR | Zbl

Cited by Sources: