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 ${\rm M} = \Gamma \backslash X$ 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 X _c , 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 X _c -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.