Slopes of adelic vector bundles over global fields. (Pentes des fibrés vectoriels adéliques sur un corps global.) (French) Zbl 1206.14047

Gaudron, Éric

Rend. Semin. Mat. Univ. Padova 119, 21-95 (2008).

This paper is a generalization of Bost’s slope theory for Hermitian vector bundles over number fields, whose importance is the application to Diophantine approximation, known as the slope method. The adelic approach in this paper provides a generalization to all global fields, which recovers Bost’s theory in the case of number fields. A possible application is seen in a slope method for function fields.
The underlying definition of the theory developed in this paper is that of an adelic vector bundle over a global field \(k\), which is vector space \(E\) over \(k\) together with a \(^{Gal}(\mathbb{C}_v,k_v)\)-invariant norm \(||\cdot ||_v:E\otimes_k \mathbb{C}_v \to \mathbb{R}_{\geq0}\) for every place \(v\) of \(k\). Here, \(\mathbb{C}_v\) denotes the completion of the algebraic closure of \(k_v\).
After an introduction, the author reviews in Section 2 some classical methods: Minkowski theory, the ellipsoids of maximal and minimal volume (“John ellipsoids” resp.“Löwner ellipsoids”), quotient volumes, the Banach-Mazur distance of two convex sets, direct sums and tensor products of normed vector spaces.
In Section 3, the author introduces adelic vector bundles and explains how to connect them to Hermitian vector bundles. He provides basic constructions as subvectorbundles, quotients, direct sums, duals, operator norms, tensor products, exterior products, determinants and symmetric products.
In Section 4, Gaudron defines the adelic degree of an adelic vector bundle \((E,\{||\cdot||_v\})\), which is the logarithmic difference of the volume of the unit ball in \(E\) with respect to all local norms \(||\cdot ||_v\) and the volume of the unit ball with respect to the quadratic norm \(||\cdot||_2\) for some choosen basis. He introduces John and Löwner vector bundles, respectively, in the adelic context and investigates their adelic degrees. He eplains how the adelic degree behaves with respect to extensions of scalars, direct sums, duals, quotients and sums of embedded adelic vector bundles.
In section 5, he defines the slope of an adelic vector bundle, together with some other invariants. Most important among these is the maximal slope, whose existence is proved in the first part of this section. Gaudron reviews the canonical filtration of a Hermitian vector bundle from the adelic view point and discusses semi-stable vector bundles.
In section 6, the author compares the slopes of adelic vector bundles in terms of morphisms between them.
In section 7, he proves that the maximal slope of the \(l\)-th symmetric power of an adelic vector bundle \(E\) equals \(l\) times the maximal slope of \(E\).
Finally, in Section 8, Gaudron explains some possible applications to geometry.

Reviewer: Oliver Lorscheid (New York)

MSC:

14G40	Arithmetic varieties and schemes; Arakelov theory; heights
11R56	Adèle rings and groups
14D20	Algebraic moduli problems, moduli of vector bundles

Keywords:

(Hermitian) vector bundles; slope method; global field; adele ring

References:

[1]	W. BLASCHKE, Über affine Geometrie VII: Neue Extremeigenschaften von Ellipse und Ellipsoid. Ber. Vehr. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl, 69 (1917), pp. 306-318. JFM46.1112.06 · JFM 46.1112.06
[2]	E. BOMBIERI - J. VAALER, On Siegel’s lemma. Invent. Math., 73 (1) (1983), pp. 11-32, Avec un addendum : ibid. 75(2) (1984), p. 377. Zbl0533.10030 MR707346 · Zbl 0533.10030 · doi:10.1007/BF01393823
[3]	T. BOREK, Successive minima and slopes of hermitian vector bundles over number fields. J. Number Theory, 113 (2) (2005), pp. 380-388. Zbl1100.14513 MR2153282 · Zbl 1100.14513 · doi:10.1016/j.jnt.2004.12.003
[4]	J.-B. BOST - H. GILLET - C. SOULÉ, Heights of projective varieties and positive Green forms. J. Amer. Math. Soc., 7 (4) (1994), pp. 903-1027. Zbl0973.14013 MR1260106 · Zbl 0973.14013 · doi:10.2307/2152736
[5]	J.-B. BOST, Intrinsic heights of stable varieties and abelian varieties. Duke Math. J., 82 (1) (1996), pp. 21-70. Zbl0867.14010 MR1387221 · Zbl 0867.14010 · doi:10.1215/S0012-7094-96-08202-2
[6]	J.-B. BOST, Périodes et isogénies des variétés abéliennes sur les corps de nombres (d’après D. Masser et G. Wüstholz). Séminaire Bourbaki, Exp. no 795. Volume 237 d’Astérisque, pp. 115-161. Société Mathématique de France, 1996. Zbl0936.11042 MR1423622 · Zbl 0936.11042
[7]	J.-B. BOST, Algebraic leaves of algebraic foliations over number fields. Publ. Math. Inst. Hautes Études Sci, 93 (2001), pp. 161-221. Zbl1034.14010 MR1863738 · Zbl 1034.14010 · doi:10.1007/s10240-001-8191-3
[8]	N. BOURBAKI, Éléments de mathématiques. Fasc. XXX. Algèbre commutative. Chapitre 5 : Entiers. Chapitre 6 : Valuations. Actualités Scientifiques et Industrielles, No. 1308. Hermann (Paris), 1964. Zbl0205.34302 MR194450 · Zbl 0205.34302
[9]	N. BOURBAKI, Espaces vectoriels topologiques. Chapitres 1 à 5. Éléments de mathématique. Nouvelle édition. Masson, Paris, 1981. Zbl0482.46001 MR633754 · Zbl 0482.46001
[10]	J. BOURGAIN - V. MILMAN, New volume ratio properties for convex symmetric bodies in Rn . Invent. Math., 88 (2) (1987) pp. 319-340. Zbl0617.52006 MR880954 · Zbl 0617.52006 · doi:10.1007/BF01388911
[11]	A. CHAMBERT-LOIR, Points de petite hauteur sur les variétés semi-abéliennes. Ann. Sci. École Norm. Sup., 33 (6) (2000), pp. 789-821. Zbl1018.11034 MR1832991 · Zbl 1018.11034 · doi:10.1016/S0012-9593(00)01053-3
[12]	A. CHAMBERT-LOIR, Théorèmes d’algébricité en géométrie diophantienne (d’après J.-B. Bost, Y. André, D. & G. Chudnovsky). Séminaire Bourbaki, Exp. no 886. Volume 282 d’Astérisque, 175-209. Société Mathématique de France, 2002. Zbl1044.11055 MR1975179 · Zbl 1044.11055
[13]	C. CHEVALLEY, Introduction to the theory of algebraic functions of one variable. Mathematical surveys, no. VI, American Mathematical Society, 1951. Zbl0045.32301 MR42164 · Zbl 0045.32301
[14]	A. DEFANT - K. FLORET, Tensor norms and operator ideals, volume 176 de North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1993. Zbl0774.46018 MR1209438 · Zbl 0774.46018
[15]	K. FLORET, The extension theorem for norms on symmetric tensor products of normed spaces. Recent progress in functional analysis (Valencia, 2000), volume 189 de North-Holland Math. Stud., 225-237. North-Holland, 2001. Zbl1010.46019 MR1861759 · Zbl 1010.46019
[16]	É. GAUDRON, Formes linéaires de logarithmes effectives sur les variétés abéliennes. Ann. Sci. École Norm. Sup., 39 (5) (2006), pp. 699-773. Zbl1111.11038 MR2292632 · Zbl 1111.11038 · doi:10.1016/j.ansens.2006.09.001
[17]	É. GAUDRON, Étude du cas rationnel de la théorie des formes linéaires de logarithmes. J. Number Theory, 127 (2) (2007), pp. 220-261. Zblpre05231962 MR2362434 · Zbl 1197.11085
[18]	P. GRAFTIEAUX, Formal groups and the isogeny theorem. Duke Math. J., 106 (1) (2001) pp. 81-121. Zbl1064.14045 MR1810367 · Zbl 1064.14045 · doi:10.1215/S0012-7094-01-10614-5
[19]	P. GRAFTIEAUX, Formal subgroups of abelian varieties. Invent. Math., 145 (1) (2001), pp. 1-17. Zbl1064.14047 MR1839283 · Zbl 1064.14047 · doi:10.1007/s002220100137
[20]	S. LANG, Algebraic number theory, volume 110 de Graduate Texts in Mathematics, Springer-Verlag, N. Y., 1994. Zbl0811.11001 MR1282723 · Zbl 0811.11001
[21]	K. MAHLER, Ein Übertragungsprinzip für konvexe Körper. Casopis Pest. Mat. Fys., 68 (1939), pp. 93-102,. MR1242 JFM65.0175.02 · Zbl 0021.10403
[22]	V. MAILLOT, Géométrie d’Arakelov des variétés toriques et fibrés en droites intégrables, volume 80 de Mémoire de la société mathématique de France, S. M. F., 2000. Zbl0963.14009 · Zbl 0963.14009
[23]	G. PISIER, The volume of convex bodies and Banach space geometry, volume 94 de Cambridge Tracts in Mathematics, Cambridge University press, 1989. Zbl0698.46008 MR1036275 · Zbl 0698.46008
[24]	H. RANDRIAMBOLOLONA, Hauteurs pour les sous-schémas et exemples d’utilisation de méthodes arakeloviennes en théorie de l’approximation diophantienne, Thèse de doctorat, Université Paris XI (Orsay), janvier 2002.
[25]	R. REMMERT, Classical topics in complex function theory, volume 172 de Graduate Texts in Mathematics, Springer-Verlag, N. Y., 1998. Zbl0895.30001 MR1483074 · Zbl 0895.30001
[26]	M. ROGALSKI, Sur le quotient volumique d’un espace de dimension finie, Séminaire Initiation à l’Analyse, G. Choquet, M. Rogalski, J. Saint-Raymond, 20e année, 1980/81, no. 3. Volume 46 de Publ. Math. univ. Pierre et Marie Curie, Paris VI. Zbl0517.46014 MR670807 · Zbl 0517.46014
[27]	D. ROY - J.L. THUNDER, An absolute Siegel’s lemma. J. Reine angew. Math., 476 (1996), pp. 1-26. Addendum et erratum, ibid., 508 (1999), 47-51. Zbl0910.11028 · Zbl 0910.11028 · doi:10.1515/crll.1999.508.47
[28]	R. RUMELY - C.F. LAU - R. VARLEY, Existence of the sectional capacity, volume 145, n. 690, de Memoirs of the American Mathematical Society, A.M.S., 2000. Zbl0987.14018 MR1677934 · Zbl 0987.14018
[29]	R.A. RYAN, Introduction to tensor products of Banach spaces. Springer Monographs in Math. Springer-Verlag London Ltd., 2002. Zbl1090.46001 MR1888309 · Zbl 1090.46001
[30]	J. SAINT-RAYMOND, Sur le volume des corps convexes symétriques, Séminaire Initiation à l’Analyse, G. Choquet, M. Rogalski, J. Saint-Raymond, 20e année, 1980/81, no. 11. Volume 46 de Publ. Math. univ. Pierre et Marie Curie, Paris VI. Zbl0531.52006 MR670798 · Zbl 0531.52006
[31]	L.A. SANTALÓ, Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugal Math., 8 (1949), pp. 155-161. Zbl0038.35702 · Zbl 0038.35702
[32]	W.M. SCHMIDT, A remark on the heights of subspaces. A tribute to Paul Erdös, pp. 359-360. Cambridge Univ. Press, 1990. Zbl0714.11022 MR1117028 · Zbl 0714.11022
[33]	R. SCHNEIDER, Convex bodies: the Brunn-Minkowski theory, volume 44 de Enyclopedia of mathematics and its applications, Cambridge University Press, 1993. Zbl0798.52001 MR1216521 · Zbl 0798.52001
[34]	C. SOULÉ, Successive minima on arithmetic varieties. Compositio Math., 96 (1) (1995), pp. 85-98. Zbl0871.14019 MR1323726 · Zbl 0871.14019
[35]	T. STRUPPECK - J.D. VAALER, Inequalities for heights of algebraic subspaces and the Thue-Siegel principle. Analytic number theory (Allerton Park, IL, 1989), volume 85 de Progress in Math., 493-528. Birkhäuser Boston, 1990. Zbl0722.11033 MR1084199 · Zbl 0722.11033
[36]	L. SZPIRO, Degrés, intersections, hauteurs. Séminaire sur les pinceaux arithmétiques : la conjecture de Mordell, volume 127 d’Astérique, 11-28. Société Mathématique de France, 1985. MR801917
[37]	A.C. THOMPSON, Minkowski geometry, volume 63 de Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1996. Zbl0868.52001 MR1406315 · Zbl 0868.52001
[38]	J.L. THUNDER, An adelic Minkowski-Hlawka theorem and an application to Siegel’s lemma. J. reine angew Math., 475 (1996), pp. 167-185. Zbl0858.11034 MR1396731 · Zbl 0858.11034 · doi:10.1515/crll.1996.475.167
[39]	E. VIADA, Slopes and abelian subvariety theorem. J. Number Theory, 112 (1) (2005) pp. 67-115. Zbl1080.11048 MR2131141 · Zbl 1080.11048 · doi:10.1016/j.jnt.2004.07.015
[40]	A. WEIL, Basic number theory, Seconde édition de 1973 publiée dans Classics in Mathematics, Springer-Verlag, Berlin, 1995. Zbl0823.11001 MR1344916 · Zbl 0823.11001
[41]	S. ZHANG, Positive line bundles on arithmetic varieties. J. Amer. Math. Soc., 8 (1) (1995), pp. 187-221. Zbl0861.14018 MR1254133 · Zbl 0861.14018 · doi:10.2307/2152886