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 $G_{\mathbb{R}}$-invariant complex structure on any non-classical period domain $D=G_{\mathbb{R}}\mathrm{/}V$ with $G_{\mathbb{R}}$ 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.