Singular holomorphic foliations with attractive leaves

Abstract

The article starts with a survey on coherent singular holomorphic foliations on connected complex manifolds. These foliations are studied using techniques of dynamical systems, notions like limit sets, basins of attraction, attractors being defined geometrically. Under the assumptions that there are leaves everywhere, that there exist only ”few” compact leaves and that the space of non-compact leaves is hausdorff respectively weakly hausdorff, it is shown: The domain of attractivity of a compact leaf L is an analytic subvariety of dimension bigger than the dimension of the leaves; it consists of the union of L with the whole space of non-compact leaves, if L is almost attractive; in case the foliation defines an open equivalence relation, L is always attractive. For every non-compact leaf L the number of almost global attractors is bounded from above by the number of connected components of the limit set of L, which again is bounded from above by the number of ends of L. These results are interpreted for holomorphic C-actions. The associated holomorphic foliations are mostly singular, on IP^n for example they are always singular. The general theory is illustrated by diagonal C-actions on IP^n of general type.

Keywords

• singular holomorphic foliations
• few compact leaves
• almost global attractors
• limit cycles
• ends of leaves
• holomorphic c-actions
• thullen’s singularity theorem