Projective varieties invariant by one - dimensional foliations

This work concerns the problem of relating characteristic numbers of onedimensional holomorphic foliations of PC 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 PC then, since line bundles over PC are classified by the Chern class c1(L) ∈ H (PC,Z) ≃ Z, one-dimensional holomorphic foliations F of PC are given by morphisms Φ : O(1 − d) −→ TPC with d ≥ 0, d ∈ Z, which we call the degree of F . We will use the notation F for such a foliation. Suppose now V i −→ PC is an irreducible algebraic variety invariant by F in such a way that the pull-back