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

  • B. Klingler

    Ecole Polytechnique, Palaiseau, France

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.

Cite this article

B. Klingler, Un théorème de rigidité non-métrique pour les variétés localement symétriques hermitiennes. Comment. Math. Helv. 76 (2001), no. 2, pp. 200–217

DOI 10.1007/S00014-001-8320-0