Projective varieties invariant by one-dimensional foliations
Abstract
This work concerns the problem of relating characteristic numbers of one-dimensional holomorphic foliations of Pn 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 :L -> TM where L is a holomorphic line bundle on M. The singular set of F is the analytic subvariety sing(F) = p : (p)=0 and the leaves of F are the leaves of the nonsingular foliation induced by F on M-sing(F). If M is Pn then, since line bundles over Pn are classified by the Chern class c1(L) in H2(Pn,Z) = Z, one-dimensional holomorphic foliations F of Pn are given by morphisms Phi:O(1-d) -> TPn with d >= 0, d in Z, which we call the degree of F. We will use the notation Fd for such a foliation. Suppose now i:V -> Pn is an irreducible algebraic variety invariant by Fd in such a way that the pull-back i*(Fd) of Fd 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.
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