Commentarii Mathematici Helvetici


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Volume 76, Issue 2, 2001, pp. 200–217
DOI: 10.1007/s00014-001-8320-0

Published online: 2001-06-30

Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes

B. Klingler[1]

(1) Ecole Polytechnique, Palaiseau, France

Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let $ {\rm M} = \Gamma \backslash X $ be a quotient of X by a torsion-free discrete subgroup $ \Gamma $ of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a $ (G_{\Bbb C} , X_c) $-structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group $ {\rm G}_{\Bbb C} $. By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the $ ({\rm G}_{\Bbb C} , X_c) $-structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.

Keywords: Locally symmetric spaces, rigidity, projective structures, uniformization

Klingler B.: Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes. Comment. Math. Helv. 76 (2001), 200-217. doi: 10.1007/s00014-001-8320-0