An application of a theorem of Huber in holomorphic foliation theory

Abstract

Introduction. In the following we are concerned with 1-codimensional holomorphic foliations on a connected paracompact complex manifold X of dimension n. Let U be an open subset of X and f : U→C a holomorphic submersion onto a 1-dimensional complex manifold C. f : U→C is called a regular local holomorphic foliation of codimension one. Two regular local holomorphic foliations f1 : U1 → C1, f2 : U2 → C2 are called compatible, if for every x ∈ U1 ∩ U2 there exist an open neighborhood W ⊂ U1∩U2 of x and a biholomorphic mapping g : f1(W )→ f2(W ) such that f2 = g ◦ f1 on W . A (global) regular holomorphic foliation F of codimension one on X is a system {fj : Uj → Cj : j ∈ J} of compatible regular local holomorphic foliations of codimension one such that ⋃ j∈J Uj = X. We identify two regular foliations F1,F2 on X if every local foliation of F1 is compatible with every local foliation of F2. In the following we assume that every regular foliation F on X contains every local foliation which is compatible with those of F . By a theorem of Frobenius there is a one to one correspondence between the system of regular holomorphic foliations F of codimension 1 on X and the system of subsheaves Ω′ of the sheaf Ω of holomorphic Pfaffian forms on X such that Ω1/Ω′ is a locally free O-sheaf of rank n− 1 and ω ∧ dω = 0 for every ω ∈ Ωx, x ∈ X. Let F be a regular holomorphic foliation on X of codimension 1. A subset L of X is called a local leaf or a plaque of F , if there is a local holomorphic foliation f : U → C of F such that L is a connected component of a fiber of f . The relatively open subsets of the local leaves of F constitute a base of a topology T on X. T is called the F-topology. (X, T ) is a complex manifold of dimension n − 1. It is not connected. The connected components L of (X, T ) are called