Study of moduli spaces -compactification and period map

1.我们于5月在名古屋大学举办了一次关于模空间的研讨会，邀请了Alexeev教授和Sankaran教授。我们讨论了Nakamura和Alexeev构造的主极化交换簇的模空间的典范紧化。在这项研究中，我们构造了8次极化阿贝尔曲面的模空间(分别为10次)和八面体(呼吸二十面体)3次之间的自然双调映射。我们发现，原木变差的概念使结构更加清晰，并且在阶数为8.2的情况下，一个完整的积分系统是相关的。我们通过代数群的作用定义了一个新的代数簇的商，它与线性化的选择无关。我们现在正在举例检验这一新的商数。将由几何不变理论得到的三次四次曲面的模空间M与周期域相比较，我们发现了一个很好的几何紧化候选M。这种紧化也表明K3曲面的模紧化也是很好的紧化。

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.