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On the stability of holomorphic foliations

  • Harald Holmann
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 798)

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.

Keywords

Holonomy Group Holomorphic Foliation Connected Fibre Unstable Foliation Compact Differentiable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Harald Holmann
    • 1
  1. 1.Institut de MathématiquesUniversité de FribourgFribourgSuisse

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