Study of moduli spaces -compactification and period map

1. 五月，我们如期在名古屋大学举办了一个关于模空间的研讨会，邀请了Alexeev和Sankaran教授。讨论了Nakamura和Alexeev构造的主极化阿贝尔变元模空间的正则紧化。在本研究中，我们构造了8次极化阿贝尔曲面的模空间与八面体曲面的模空间之间的自然双象映射。二十面体)的三倍。我们发现，对数变化的概念使结构更清晰，在8.2度的情况下，一个完整的积分系统是相关的。我们利用代数群的作用定义了一个新的阿尔及利亚变商，它与线性化的选择无关。我们现在用例子来检验这个新商。将由几何不变量理论得到的三次四折叠模空间M与周期域进行比较，我们发现了一个具有良好几何紧化M的候选空间。

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.