JournalscmhVol. 76 , No. 2DOI 10.1007/s00014-001-8320-0

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

  • B. Klingler

    Ecole Polytechnique, Palaiseau, France
Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes cover

Abstract

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 M=Γ\X{\rm M} = \Gamma \backslash X be a quotient of X by a torsion-free discrete subgroup Γ\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 (GC,Xc)(G_{\Bbb C} , X_c) -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group GC{\rm G}_{\Bbb C} . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the (GC,Xc)({\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.