Moduli of polarized Hodge structures

TL;DR

This paper discusses the theory of moduli spaces of polarized Hodge structures, exploring Griffiths' vision of enlarging the period domain and addressing the challenges in automorphic cohomology for non-Hermitian symmetric domains.

Contribution

It provides an exposition of the moduli of polarized Hodge structures and reformulates Griffiths' problem using classical uniformization, advancing understanding of degenerating Hodge structures.

Findings

Clarifies the structure of the period domain D.

Reformulates Griffiths' problem via Weierstrass uniformization.

Highlights open problems in automorphic cohomology for D.

Abstract

Around 1970 Griffiths introduced the moduli of polarized Hodge structures/the period domain $D$ and described a dream to enlarge $D$ to a moduli space of degenerating polarized Hodge structures. Since in general $D$ is not a Hermitian symmetric domain, he asked for the existence of a certain automorphic cohomology theory for $D$, generalizing the usual notion of automorphic forms on symmetric Hermitian domains. Since then there have been many efforts in the first part of Griffith's dream but the second part still lives in darkness. The objective of the present text is two-folded. First, we give an exposition of the subject. Second, we give another formulation of the Griffiths problem, based on the classical Weierstrass uniformization theorem.