Comparative Study of Various Fundamental Groups and Their Structure in Arithmetic and Topology

在维度3的情况下，最小模型有奇异点，每个奇异点都有称为索引的不变量。尽管在非奇异模型中会消失，但已知整个指标=篮子是一个分型不变量，也是一个变形不变量。同时，在对三维品种的分类中，需要某些品种的指标有界性。我之前讲过，当正则因子k在数值上等于0时指标是有界的。在“Q-Fano三倍的有界性”一文中，我证明了当K为负时，指标和K^3都是有界的。在“代数曲面的适度退化”一文中，我也用完全不同的方法证明了椭圆曲面退化指标的有界性。此外，在同一篇论文中，我还证明了曲面退化的最小模型的拓扑结构与半稳定退化的拓扑结构非常相似。如果X是定义在代数数域k上的代数变量，则基本群pi， (X)可以看作绝对伽罗瓦群Gal （k/k）的群扩展。Nakamura研究了X可以用π， (X)的群理论性质来表征的条件，并得到了几个结果。首先，他证明了当X是一条双曲曲线时，π, (X)作为Gal （k/k）的群扩展决定了X的唯一性直到同构。他还证明，当X为P' - 3点时，每一个作为群扩展的自同构，(X)必须由X本身的自同构引申而来。Matsumoto研究了退化黎曼曲面族中奇异元的拓扑分类；他证明了奇异曲面的拓扑类型是由非阿德利亚单形决定的，反之，对于任何代数有限型的单形，都存在一个退化的黎曼曲面族来实现该单形。

In the case of dimension 3, a minimal model has singular points and each singular point has the the invariant called the index. In spite of disappearance in a non-singular model, it is known that the whole of indices = the basket is a birational invariant and also a deformation invariant. By the way, in the calssification of 3-dimensional varieties, the boundedness of indices for some classes of varieties is needed. I have shown before that the indices are bounded when the canonical divisor k is numerically equivalent to zero. As a continuation of this, I have shown that both indices and K^3 are bounded when K is negative in the paper "Boundedness of Q-Fano threefolds". By the totally different method, I have also shown the boundesness of indices for degenerations of elliptic surfaces in the paper "Moderate degenerations of algebraic surfaces". Moreover, in the same paper, I have shown that the topology of minimal models of degenerations of surfaces are very similar to that of semistable degenerations.If X is an algebraic variety defined over an algebraic number field k, then the fundamental group pi, (X) can be considered as a group extension of the absolute Galois group Gal (k/k). Nakamura studied conditions under which X can be characterized by the group theoretical properties of pi, (X), and obtained several results. Firstly, he showed that when X is a certain hyperbolic curve, pi, (X) as a group extension of Gal (k/k) determines X uniquely up to isomorphisms. He also showed that, when X is P'minus three points, every automorphism of pi, (X) as a group extension must be induced from an automorphim of X itself.Matsumoto studied topological classification of singular fiders in degenerating families of Riemann surfaces ; he proved that topological types of singular fivers are determined by non-adelian monodromy and conversely that, for any monodromy of algebraically finite type, there exists a degenerating family of Riemann surfaces which realizes the monodromy.