  In this paper we will study the moduli spaces of log Hodge structures
introduced by Kato-Usui. This moduli space is a partial compactification of a
discrete quotient of a period domain. We treat the following 2 cases: (A) the
case where the period domain is Hermitian symmetric, (B) the case where the
Hodge structures are of the mirror quintic type. Especially we study a property
of the torsor.
