A theorem of non-metric rigidity for Hermitian locally symmetric manifolds

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 = Gamma \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 C-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(c), X-c) - structure on M, a maximal atlas of X-c -valued charts with locally constant transition maps in the complexified group G(C). By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the (G(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.