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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 seconddoi:10.1007/s00574-008-0073-6 fatcat:evp6lmdifzelrmqvp77qfjow4e