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 -

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