Variations of Hodge structure and orbits in flag varieties

TL;DR

This paper explores the geometry and representation theory of flag varieties related to Hodge structures, focusing on Griffiths--Yukawa coupling, polarized orbits, and boundary components, especially in adjoint flag varieties.

Contribution

It establishes new connections between Hodge theory and the geometry of flag varieties, including the relation of Griffiths--Yukawa coupling to lines on $G/P$ and construction of polarized orbits.

Findings

Relation of Griffiths--Yukawa coupling to lines on $G/P$

Construction of polarized $G_{bR}$--orbits in flag varieties

Computation of Hodge--theoretic boundary components

Abstract

Period domains, the classifying spaces for (pure, polarized) Hodge structures, and more generally Mumford-Tate domains, arise as open $G_{\mathbb{R}}$--orbits in flag varieties $G/P$. We investigate Hodge--theoretic aspects of the geometry and representation theory associated with these flag varieties. In particular, we relate the Griffiths--Yukawa coupling to the variety of lines on $G/P$ (under a minimal homogeneous embedding), construct a large class of polarized $G_{\mathbb{R}}$--orbits in $G/P$, and compute the associated Hodge--theoretic boundary components. An emphasis is placed throughout on adjoint flag varieties and the corresponding families of Hodge structures of levels two and four.