Algebraic leaves of algebraic foliations over number fields.
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 This solves an (unpublished) conjecture of \textit{T. Ekedahl} and \textit{N. Shepherd-Barron}.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 on ``transcendence techniques'', except that the traditional construction of auxiliary polynomials is replaced by the use of ``slopes'' for hermitian vector bundles over arithmetic curves, via the language of Arakelov geometry. This method was introduced by the author's work building on the work of Masser and Wüstholz [see \textit{J.-B. Bost}, Astérisque 237, 115--161 (1996; Zbl 0936.11042)].
scientific article; zbMATH DE number 1874409