Algebraic leaves of algebraic foliations over number fields

Jean-Benoît Bost[1]

  • [1] Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France.

Publications Mathématiques de l'IHÉS (2001)

  • Volume: 93, page 161-221
  • ISSN: 0073-8301

Abstract

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We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C , a smooth algebraic variety X over K , equipped with a K - rational point P , and F an algebraic subbundle of the its tangent bundle T X , defined over K . Assume moreover that the vector bundle F is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold X ( C ) , and one may consider its leaf F through P . We prove that F is algebraic if the following local conditions are satisfied: i) For almost every prime ideal p of the ring of integers 𝒪 K of the number field K, the p -curvature of the reduction modulo p of the involutive bundle F vanishes at P (where p denotes the characteristic of the residue field O K/p). ii) The analytic manifold F satisfies the Liouville property; this arises, in particular, if F is the image by some holomorphic map of the complement in a complex algebraic variety of a closed analytic subset. This algebraicity criterion unifies and extends various results of D. V. and G. V. Chudnovsky, André, and Graftieaux, and also admits new consequences. For instance, applied to an algebraic group G over K, it shows that a K-Lie subalgebra h of Lie G is algebraic if and only if for almost every non-zero prime ideal p of O K, of residue characteristic p, the reduction modulo p of h is a restricted Lie subalgebra of the reduction modulo p of Lie G (i.e., is stable under p-th powers). This solves a conjecture of Ekedahl and Shepherd-Barron. The algebraicity criterion above follows from a more basic algebraicity criterion concerning smooth formal germs in algebraic varieties over number fields. The proof of the latter relies on “transcendence techniques”, recast in a modern geometric version involving elementary concepts of Arakelov geometry, and on some analytic estimates, related to the First Main Theorem of higher-dimensional Nevanlinna theory.

How to cite

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Bost, Jean-Benoît. "Algebraic leaves of algebraic foliations over number fields." Publications Mathématiques de l'IHÉS 93 (2001): 161-221. <http://eudml.org/doc/104175>.

@article{Bost2001,
abstract = {We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field $K$ embedded in $C$, a smooth algebraic variety $X$ over $K$, equipped with a $K-$rational point $P$, and $F$ an algebraic subbundle of the its tangent bundle $T_X$, defined over $K$. Assume moreover that the vector bundle $F$ is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold $X(C)$, and one may consider its leaf $F$ through $P$. We prove that F is algebraic if the following local conditions are satisfied: i) For almost every prime ideal $p$ of the ring of integers $\mathcal \{O\} K$ of the number field K, the $p$-curvature of the reduction modulo $p$ of the involutive bundle F vanishes at P (where p denotes the characteristic of the residue field O K/p). ii) The analytic manifold F satisfies the Liouville property; this arises, in particular, if F is the image by some holomorphic map of the complement in a complex algebraic variety of a closed analytic subset. This algebraicity criterion unifies and extends various results of D. V. and G. V. Chudnovsky, André, and Graftieaux, and also admits new consequences. For instance, applied to an algebraic group G over K, it shows that a K-Lie subalgebra h of Lie G is algebraic if and only if for almost every non-zero prime ideal p of O K, of residue characteristic p, the reduction modulo p of h is a restricted Lie subalgebra of the reduction modulo p of Lie G (i.e., is stable under p-th powers). This solves a conjecture of Ekedahl and Shepherd-Barron. The algebraicity criterion above follows from a more basic algebraicity criterion concerning smooth formal germs in algebraic varieties over number fields. The proof of the latter relies on “transcendence techniques”, recast in a modern geometric version involving elementary concepts of Arakelov geometry, and on some analytic estimates, related to the First Main Theorem of higher-dimensional Nevanlinna theory.},
affiliation = {Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay cedex, France.},
author = {Bost, Jean-Benoît},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {algebraicity; foliation; Arakelov geometry; -curvature; slope},
language = {eng},
pages = {161-221},
publisher = {Institut des Hautes Etudes Scientifiques},
title = {Algebraic leaves of algebraic foliations over number fields},
url = {http://eudml.org/doc/104175},
volume = {93},
year = {2001},
}

TY - JOUR
AU - Bost, Jean-Benoît
TI - Algebraic leaves of algebraic foliations over number fields
JO - Publications Mathématiques de l'IHÉS
PY - 2001
PB - Institut des Hautes Etudes Scientifiques
VL - 93
SP - 161
EP - 221
AB - We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field $K$ embedded in $C$, a smooth algebraic variety $X$ over $K$, equipped with a $K-$rational point $P$, and $F$ an algebraic subbundle of the its tangent bundle $T_X$, defined over $K$. Assume moreover that the vector bundle $F$ is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold $X(C)$, and one may consider its leaf $F$ through $P$. We prove that F is algebraic if the following local conditions are satisfied: i) For almost every prime ideal $p$ of the ring of integers $\mathcal {O} K$ of the number field K, the $p$-curvature of the reduction modulo $p$ of the involutive bundle F vanishes at P (where p denotes the characteristic of the residue field O K/p). ii) The analytic manifold F satisfies the Liouville property; this arises, in particular, if F is the image by some holomorphic map of the complement in a complex algebraic variety of a closed analytic subset. This algebraicity criterion unifies and extends various results of D. V. and G. V. Chudnovsky, André, and Graftieaux, and also admits new consequences. For instance, applied to an algebraic group G over K, it shows that a K-Lie subalgebra h of Lie G is algebraic if and only if for almost every non-zero prime ideal p of O K, of residue characteristic p, the reduction modulo p of h is a restricted Lie subalgebra of the reduction modulo p of Lie G (i.e., is stable under p-th powers). This solves a conjecture of Ekedahl and Shepherd-Barron. The algebraicity criterion above follows from a more basic algebraicity criterion concerning smooth formal germs in algebraic varieties over number fields. The proof of the latter relies on “transcendence techniques”, recast in a modern geometric version involving elementary concepts of Arakelov geometry, and on some analytic estimates, related to the First Main Theorem of higher-dimensional Nevanlinna theory.
LA - eng
KW - algebraicity; foliation; Arakelov geometry; -curvature; slope
UR - http://eudml.org/doc/104175
ER -

References

top
  1. [Anc90] A. ANCONA, Théorie du potentiel sur les graphes et les variétés, in École d’été de Probabilités de Saint-Flour XVIII-1988, Lectures Notes in Mathematics 1427, pages 1-112, Berlin, Springer, 1990. Zbl0719.60074
  2. [And89] Y. ANDRÉ, G-functions and geometry, Braunschweig, Friedr. Vieweg & Sohn, 1989. Zbl0688.10032MR990016
  3. [And99] Y. ANDRÉ, Sur la conjecture des p-courbures de Grothendieck et Katz, Preprint, Institut de Mathématiques de Jussieu, 1999. 
  4. [AS60] L. V. AHLFORS and L. SARIO, Riemann surfaces, Princeton, N.J., Princeton University Press, 1960, Princeton Mathematical Series, No. 26. Zbl0196.33801MR114911
  5. [BMQ01] F. A. BOGOMOLOV and M. L. MCQUILLAN, Rational curves on foliated varieties, Preprint, IHES, 2001. Zbl06584116
  6. [Bom81] E. BOMBIERI, On G-functions, in Recent progress in analytic number theory, Vol. 2 (Durham, 1979), pages 1-67. London, Academic Press, 1981. Zbl0461.10031MR637359
  7. [Bos96] J.-B. BOST, Périodes et isogénies des variétés abéliennes sur les corps de nombres (d’après D. Masser et G. Wüstholz), Séminaire Bourbaki, 1994 1995, Exposé N° 795, Astérisque, 237 (1996), 115-161. Zbl0936.11042
  8. [Bos98] J.-B. BOST, µmax(SkE) kE) k[µmax(E) + C(rk E)], Letter to P. Graftieaux, December 1998. 
  9. [BGS94] J.-B. BOST, H. GILLET, and C. SOULÉ, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc., 7 (1994), 903-1027. Zbl0973.14013MR1260106
  10. [CB01] A. CHAMBERT-LOIR, Théorèmes d’algébricité en géométrie diophantienne (d’après J.-B. Bost, Y. André, D. et G. Chudnovsky), Séminaire Bourbaki, Exposé 886, mars 2001. Zbl1044.11055
  11. [CC85a] D. V. CHUDNOVSKY and G. V. CHUDNOVSKY, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, in Number theory (New York, 1983-84), Lectures Notes in Mathematics 1135, pages 52-100, Berlin, Springer, 1985. Zbl0565.14010MR803350
  12. [CC85b] D. V. CHUDNOVSKY and G. V. CHUDNOVSKY, Padé approximations and Diophantine geometry, Proc. Nat. Acad. Sci. U.S.A., 82 (1985), 2212-2216. Zbl0577.14034MR788857
  13. [DGS94] B. DWORK, G. GEROTTO, and F. J. SULLIVAN, An introduction to G-functions, Princeton, N.J., Princeton, University Press, 1994. Zbl0830.12004MR1274045
  14. [Eke87] T. EKEDAHL, Foliations and inseparable morphisms, in Algebraic geometry - Bowdoin 1985, Proc. Symp. Pure Math. 46-2, pages 139-149, Amer. Math. Soc., Providence, RI, 1987. Zbl0659.14018MR927978
  15. [ESB99] T. EKEDAHL and N. I SHEPHERD-BARRON, A conjecture on the existence of compact leaves of algebraic foliations, Preprint, April 1999. 
  16. [Fal83] G. FALTINGS, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., 73 (1983), 349-366. Zbl0588.14026MR718935
  17. [Gr98] P. GRAFTIEAUX, Groupes formels et critères d’isogénie, Thèse, Université Paris VI, March 1998. 
  18. [Gr01a] P. GRAFTIEAUX, Formal groups and the isogeny theorem, Duke Math. J., 106 (2001), 81-121. Zbl1064.14045MR1810367
  19. [Gr01b] P. GRAFTIEAUX, Formal subgroups of abelian varieties, Invent. Math., 145 (2001), 1-17. Zbl1064.14047MR1839283
  20. [GK73] P. GRIFFITHS and J. KING, Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math., 130 (1973), 145-220. Zbl0258.32009MR427690
  21. [Gri99] A. GRIGOR’YAN, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135-249. Zbl0927.58019
  22. [Gro71] A. GROTHENDIECK, Revêtements Étales et Groupe Fondamental, S.G.A.1, Lecture Notes in Mathematics 224, Berlin, Springer-Verlag, 1971. MR354651
  23. [GS92] H. GILLET and C. SOULÉ, An arithmetic Riemann-Roch theorem, Invent. Math., 110 (1992), 473-543. Zbl0777.14008MR1189489
  24. [Har68] R. HARTSHORNE, Cohomological dimension of algebraic varieties, Ann. Math., 88 (1968), 403-450. Zbl0169.23302MR232780
  25. [Hir68] H. HIRONAKA, On some formal imbeddings, Illinois J. Math., 12 (1968), 587-812. Zbl0169.52302MR241433
  26. [HM68] H. HIRONAKA and H. MATSUMURA, Formal functions and formal embeddings, J. Math. Soc. Japan, 20 (1968), 52-82. Zbl0157.27701MR251043
  27. [Hon68] T. HONDA, Formal groups and zeta-functions, Osaka J. Math., 5 (1968), 199-213. Zbl0169.37601MR249438
  28. [Hör94] L. HÖRMANDER, Notions of convexity, Boston, MA, Birkhäuser Boston Inc., 1994. Zbl0835.32001MR1301332
  29. [Kat70] N. KATZ, Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publ. Math. IHES, 39 (1970), 175-232. Zbl0221.14007MR291177
  30. [Kat72] N. KATZ, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math., 18 (1972), 1-118. Zbl0278.14004MR337959
  31. [Kat73] N. KATZ, Exposé XXII: Une formule de congruence pour la fonction , in Groupes de monodromie en géométrie algébrique II, SGA 7 II, Lecture Notes in Mathematics 340, pages 401-438. Berlin, Springer-Verlag, 1973. Zbl0275.14015MR354657
  32. [Kat82] N. KATZ, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France, 110 (1982), 203-239 and 347-348. Zbl0504.12022MR667751
  33. [KM85] N. KATZ and B. MAZUR, Arithmetic moduli of elliptic curves, Princeton, N.J., Princeton University Press, 1985. Zbl0576.14026MR772569
  34. [Kli91] M. KLIMEK, Pluripotential theory, New York, The Clarendon Press - Oxford University Press, 1991. Zbl0742.31001MR1150978
  35. [KN63] S. KOBAYASHI and K. NOMIZU, Foundations of Differential Geometry, New York, John Wiley & Sons, 1963. Zbl0175.48504
  36. [Kro80] L. KRONECKER, Über die Irreductibilität von Gleichungen, Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, pages 155-162, 1880. JFM12.0065.02
  37. [Miy87] Y. MIYAOKA, Deformations of a morphism along a foliation and applications, in Algebraic geometry-Bowdoin 1985, Proc. Symp. Pure Math. 46-1, pages 245-268. Amer. Math. Soc., Providence, RI, 1987. Zbl0659.14008MR927960
  38. [MP97] Y. MIYAOKA and T. PETERNELL, Geometry of higher-dimensional algebraic varieties, Basel, Birkhäuser Verlag, 1997. Zbl0865.14018MR1468476
  39. [Mum70] D. MUMFORD, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Bombay, 1970. Zbl0223.14022MR282985
  40. [Ser58] J.-P. SERRE, Espaces fibrés algébriques, in Séminaire C. Chevalley, Secrétariat Mathématique, Paris, 1958. 
  41. [Ser59] J.-P. SERRE, Groupes algébriques et corps de classes, Paris, Herman, 1959. Zbl0718.14001MR103191
  42. [Ser68] J.-P. SERRE, Abelian l-adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Zbl0186.25701MR263823
  43. [Sha85] B. V. SHABAT, Distribution of values of holomorphic mappings, American Mathematical Society, Providence, RI, 1985. Zbl0564.32016MR807367
  44. [SB92] N. SHEPHERD-BARRON, Miyaoka’s theorems on the generic seminegativity of TX and on the Kodaira dimension of minimal regular threefolds, Astérisque, 211 (1992), 103-114. Zbl0809.14034
  45. [Shi75] B. SHIFFMAN, Nevanlinna defect relations for singular divisors, Invent. Math., 31 (1975), 155-182. Zbl0436.32022MR430325
  46. [Sie29] C. L. SIEGEL, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss., 1, 1929. Zbl56.0180.05JFM56.0180.05
  47. [Sou97] C. SOULÉ, Hermitian vector bundles on arithmetic varieties, in Algebraic geometry-Santa Cruz 1995, Proc. Symp. Pure Math. 62-1, pages 383-419, Amer. Math. Soc., Providence, RI, 1997. Zbl0926.14011MR1492529
  48. [Sto77] W. STOLL, Aspects of value distribution theory in several complex variables, Bull. Amer. Math. Soc., 83 (1977), 166-183. Zbl0344.32015MR427692
  49. [Szp85] L. SZPIRO, Degrés, intersections, hauteurs, Astérisque, 127 (1985), 11-28. MR801917
  50. [Tak93] K. TAKEGOSHI, A Liouville theorem on an analytic space, J. Math. Soc. Japan, 45 (1993), 301-311. Zbl0788.32004MR1206655
  51. [Zha98] S.-W. ZHANG, Small points and Arakelov theory, in Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), pages 217-225, 1998. Zbl0912.14008MR1648072

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