Cycle transversal flag domains of Hodge type

TL;DR

This paper classifies all proper, equivariant embeddings of flag domains of Hodge type into period domains that satisfy a transversality condition, showing they are Hermitian symmetric spaces of classical non-compact type.

Contribution

It provides a complete classification of such embeddings, demonstrating they must be Hermitian symmetric spaces of classical non-compact type.

Findings

All proper, equivariant embeddings of Hodge type flag domains are Hermitian symmetric spaces of classical type.

The classification applies to embeddings satisfying Griffiths transversality.

Embeddings are characterized by a transversality condition in period domains.

Abstract

Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexification. A special class of flag domains constitute the classifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this paper we classify all proper, equivariant embeddings of a flag domain of Hodge type, required to satisfy a certain transversality condition, in a period domain and show that any such flag domain must be a Hermitian symmetric space of non-compact type of classical type. In particular, the classification describes also the proper, equivariant embeddings of a flag domain of Hodge type in a period domain satisfying Griffiths transversality.