Projective varieties invariant by one-dimensional foliations

Abstract

This work concerns the problem of relating characteristic numbers of one-dimensional holomorphic foliations of P^n to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a one-dimensional holomorphic foliation F of M is a morphism Φ:L -> TM where L is a holomorphic line bundle on M. The singular set of F is the analytic subvariety sing(F) = {p : Φ(p)=0} and the leaves of F are the leaves of the nonsingular foliation induced by F on M-sing(F). If M is P^n then, since line bundles over P^n are classified by the Chern class c_1(L) in H^2(P^n,Z) = Z, one-dimensional holomorphic foliations F of P^n are given by morphisms Phi:O(1-d) -> TP^n with d >= 0, d in Z, which we call the degree of F. We will use the notation F^d for such a foliation. Suppose now i:V -> P^n is an irreducible algebraic variety invariant by F^d in such a way that the pull-back i^*(F^d) of F^d to V has a finite set of points as the singular set. The problem we address is the relation between d and the degree of V.