Stability of complex foliations transverse to fibrations

Abstract:

We prove that a holomorphic foliation of codimension $k$ which is transverse to the fibers of a fibration and has a compact leaf with finite holonomy group is a Seifert fibration, i.e., has all leaves compact with finite holonomy. This is the case for $C^1$-small deformations of a foliation where the original foliation exhibits a compact leaf and the base $B$ of the fibration satisfies $H^1(B,\mathbb R)=0$ and $H^1(B, \operatorname {GL}(k,\mathbb R))=0$.