%0 Journal Article
%T A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations
%A Bruno Scardua
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/585298
%X We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset. 1. Introduction Foliations transverse to fibrations are among the very first and simplest constructible examples of foliations, accompanied by a well-known transverse structure. These foliations are suspensions of groups of diffeomorphisms and their behavior is closely related to the action of the group in the fiber. For these reasons, many results holding for foliations in a more general context are first established for suspensions, that is, foliations transverse to a fibration. In this paper, we pursue this idea, but not restricted to it. We investigate versions of the classical stability theorems of Reeb [1, 2], regarding the behavior of the foliation in a neighborhood of a compact leaf, replacing the finiteness of the holonomy group of the leaf by the existence of a sufficient number of compact leaves. This is done for transversely holomorphic (or transversely analytic) foliations. Let be a (locally trivial) fibration with total space , fiber , base , and projection . A foliation on is transverse to if: (1) for each , the leaf of with is transverse to the fiber , ; (2) ; (3) for each leaf of , the restriction is a covering map. A theorem of Ehresmann ([1] Chpter V) [2]) assures that if the fiber is compact, then conditions (1) and (2) together already imply (3). Such foliations are conjugate to suspensions and are characterized by their global holonomy ([1], Theorem 3, page 103 and [2], Theorem 6.1, page 59). The codimension one case is studied in [3]. In [4], we study the case where the ambient manifold is a hyperbolic complex manifold. In [5], the authors prove a natural version of the stability theorem of Reeb for (transversely holomorphic) foliations transverse to fibrations. A foliation on is called a Seifert fibration if all leaves are compact with finite holonomy groups. The following stability theorem is proved in [5]. Theorem 1.1. Let be a holomorphic foliation transverse to a fibration with fiber . If has a compact leaf with finite holonomy group then is a Seifert fibration. It is also observed in [5] that the existence of a trivial holonomy compact leaf is assured if is of codimension has a compact leaf, and the base satisfies . Since a foliation transverse to a
%U http://www.hindawi.com/journals/ijde/2012/585298/