Compact Quotients Of Non-Classical Domains Are Not Kahler

Let D = G/V be a non-classical period domain, where G is a semi-simple real Lie group with a compact Cartan subgroup T and maximal compact subgroup K, and V the centralizer of a torus in G. If the symmetric space is Hermitian, assume also that G/V does not fiber holomorphically over G/K. Let Gamma be a co-compact, torsion-free lattice in G. We prove that the complex manifold Gamma\G/V does not admit any Kahler metric.