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Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

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Abstract

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group.

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Authors and Affiliations

  1. Département de Mathématiques d’Orsay, équipe de Topologie et Dynamique, Bat. 425, UMR 8628 CNRS, Univ. Paris-Sud (11), 91405, Orsay Cedex, France

    Sorin Dumitrescu

  2. CNRS, UMPA, école Normale Supérieure de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France

    Abdelghani Zeghib

Authors
  1. Sorin Dumitrescu
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  2. Abdelghani Zeghib
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Correspondence to Sorin Dumitrescu.

Additional information

S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.

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Dumitrescu, S., Zeghib, A. Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds. Math. Ann. 345, 53–81 (2009). https://doi.org/10.1007/s00208-009-0342-8

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