Preprint A575/2007
Moduli of Polarized Hodge Structures

Hossein Movasati

**Keywords: **
Hodge structure

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.