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

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 $\mathrm{M}=\Gamma \backslash X$ 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 $(G_{\mathbb{C}},X_c)$ -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group ${\mathrm{G}}_{\mathbb{C}}$ . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the $({\mathrm{G}}_{\mathbb{C}},X_c)$ -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.