On the stability of holomorphic foliations

Abstract

In 1978 D.B.A. Epstein and E. Vogt succeeded in constructing an unstable real-analytic periodic flow on a 4-dimensional compact real-analytic manifold. This cannot be generalized to the complex-analytic case:

Proposition 1: A periodic holomorphic flow of a compact complex variety is always stable (H. Holmann, 1977).

This year Th. Müller found the first example of an unstable compact holomorphic foliation of a non-compact complex manifold in form of a periodic holomorphic flow all orbits being equivalent complex tori. The underlying 3-dimensional complex manifold of this example cannot carry a Kähler structure because of the following proposition proved in this paper:

Proposition 2: On a (not necessarily compact) Kähler manifold all compact holomorphic foliations are stable.

Its proof uses that Kähler-manifolds are characterized by the fact that local-analytic submanifolds are minimal surfaces with respect to the Kähler metric. Proposition 2 is a special case of a more general result obtained with different methods by H. Rummler (1978):

Proposition 3: A compact differentiable foliation of a differentiable manifold is stable iff it carries a Riemannian-metric such that all leaves are minimal surfaces with respect to this metric.