Study of moduli spaces -compactification and period map

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1. We held a workshop on the moduli space in May as sheduled in NagoyaUniversity invitiing Profeeeors Alexeev and Sankaran. We discussed the canonical compactification of the moduli space of principally polarized abelian varieties constructed by Nakamura and Alexeev. In this study, we have constructed natural birational map between the moduli space of polarized abelian surfaces of degree 8 (resp.10) and the octahedral (resp.icosahedral) 3-fold. We found that the notion of log variety makes the construction clearer and that a complete integral system is related in the case of degree 8.2. We defined a new quotient of algerbaic variety by an action of an algebraic group which is independent of the choice of a linearization. We are now examining this new quotient in examples.3. Comparing the moduli space M of cubic 4-folds obtained from the geometric invariant theory with the period domain, we found a candidate for good geometric compactification M.This compactification suggests a good compactification of the moduli for K3 surfaces also.