Algebraic leaves of algebraic foliations over number fields.
type: Item
instance of: scholarly article
title: Algebraic leaves of algebraic foliations over number fields.
zbMATH Open document ID: 1034.14010
author: Jean-Benoît Bost
published in: Publications Mathématiques
publication date: 2001-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104175
review text: This paper proves an algebraicity criterion for leaves of algebraic foliations over number fields. Let \(K\) be a number field embedded in \(\mathbb C\), let \(X\) be a smooth algebraic variety over \(K\) (i.e., an integral separated scheme of finite type over \(K\)), and let \(F\) be an algebraic subbundle of the tangent bundle \(T_X\). We assume that \(F\) is involutive; i.e., closed under the Lie bracket. Then \(F\) defines a holomorphic foliation of the complex manifold \(X(\mathbb C)\). This paper proves that the leaf \(\mathcal F\) through a rational point \(P\in X(K)\) is algebraic if the following local conditions are satisfied: (i) for almost all prime ideals \(\mathfrak p\) of the ring of integers \({\mathfrak O}_K\) of \(K\), the \(p\)-curvature of the reduction modulo \(\mathfrak p\) of the subbundle \(F\subseteq T_X\) vanishes at \(P\) (here \(p\) is the prime of \(\mathbb Z\) lying below \(\mathfrak p\)); and (ii) the manifold \(\mathcal F\) satisfies the Liouville property: Every plurisubharmonic function on \(\mathcal F\) bounded from above is constant. [For example, \(\mathcal F\) satisfies the Liouville property if it is a holomorphic image of a complex algebraic variety minus a closed analytic subset.] These conditions are (in the words of the author) ``almost necessary'', in the sense that (ii) is necessary, and for almost all \(\mathfrak p\), the \(p\)-curvature vanishes at the point where \(P\) meets the closed fiber at \(\mathfrak p\). As an application, let \(G\) be an algebraic group over \(K\). Then a \(K\)-Lie subalgebra \(\mathbf h\) of \(\text{Lie\,} G\) is algebraic if and only if for almost all primes \(\mathfrak p\) of \(\mathcal O_K\), the reduction modulo \(\mathfrak p\) of \(\mathbf h\) is a restricted Lie subalgebra of the reduction modulo \(\mathfrak p\) of \(\text{Lie\,} G\) (i.e., it is stable under \(p\)-th powers). This solves an (unpublished) conjecture of \textit{T. Ekedahl} and \textit{N. Shepherd-Barron}. The proof relies…
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G35
Mathematics Subject Classification ID: 14G25
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 1874409
zbMATH Keywords: algebraicity
zbMATH Keywords: foliation
zbMATH Keywords: Arakelov geometry
zbMATH Keywords: \(p\)-curvature
zbMATH Keywords: slope
reviewed by: Paul Vojta
MaRDI profile type: Publication
OpenAlex ID: W2067894604
DOI: 10.1007/S10240-001-8191-3
description: scientific article; zbMATH DE number 1874409