Geometry of non-classical period domains

In this paper we prove a conjecture of Griffiths about vanishing of the zeroth cohomology groups of locally homogeneous vector bundles on compact quotients of non-classical period domains, and construct a new G_-invariant complex structure on any non-classical period domain D=G_/V with G_ of Hermitian type. Various geometric and algebraic characterizations of non-classical period domains and several geometric applications on their compact quotients are deduced as consequences of our results.