Geometry of non-classical period domains

TL;DR

This paper proves Griffiths' conjecture on cohomology vanishing for certain vector bundles on non-classical period domains and introduces a new invariant complex structure, with applications to their geometry.

Contribution

It establishes a vanishing theorem for cohomology groups and constructs a novel invariant complex structure on non-classical period domains.

Findings

Proved Griffiths' conjecture on cohomology vanishing.

Constructed a new $G_ $-invariant complex structure.

Derived geometric characterizations and applications.

Abstract

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