IJPAM: Volume 4, No. 3 (2003)

STABILITY OF 1-CODIMENSIONAL
ANALYTIC DECOMPOSITIONS

H. Holmann

, B. Kaup

, H.-J. Reiffen

$^{1,2}$ Department of Mathematics
University of Fribourg Suisse
Chemin du Musée 23, CH-1700 Fribourg, SWITZERLAND

e-mail: harald.holmann@unifr.ch

e-mail: burchard.kaup@unifr.ch

Department of Mathematics and Informatics
University of Osnabrück
D-49069 Osnabrück, GERMANY
e-mail: Reiffen@mathematik.uni-osnabrueck.de

Abstract.We define an analytic decomposition of a complex manifold of dimension to be an equivalence relation $\D$ such that all classes (we call them leaves) are connected analytic subsets of pure codimension one and such that the sheaf of vector fields, which are tangent to all classes, is coherent and has rank . Such decompositions occur in a natural way as systems of leaves of certain singular holomorphic foliations. We give sufficient conditions, under which $\D$ is stable in the following sense: for every leaf and for every compact subset $C \subset X \setminus L$ there exists an open saturated neighborhood of satisfying $C \cap U = \emptyset$ . In particular, if all leaves are compact, then $\D$ is stable iff $X/\D$ is hausdorff iff $X/\D$ is a Riemann surface iff $\D \subset X \times X$ is analytic.