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

在3维的情况下，最小模型具有奇异点，并且每个奇异点具有称为索引的不变量。尽管消失在一个非奇异模型，它是已知的，整个指数=篮子是一个双有理不变量，也是一个变形不变量。另外，在三维簇的分类中，某些簇的指数是有界的。我以前已经证明，当标准因子k在数值上等价于零时，指数是有界的。作为这一点的延续，我在论文“Q-Fano threefolds的有界性”中证明了当K为负时，两个指数和K^3都有界。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，在同一篇文章中，我还证明了曲面退化的极小模型的拓扑与半稳定退化的拓扑非常相似，如果X是定义在代数数域k上的代数簇，则基本群pi（X）可以看作是绝对Galois群Gal（k/k）的群扩张。中村研究的条件下，X可以表征的群论性质的pi，（X），并获得了几个结果。首先，他表明，当X是一个特定的双曲曲线，π，（X）作为一个组的扩展Gal（k/k）确定X唯一的同构。他还证明了当X是P '减三点时，π（X）作为群扩张的每一个自同构都必须由X自身的一个自同构导出.松本研究了退化黎曼曲面族中奇异fiders的拓扑分类;他证明了奇异fivers的拓扑类型是由非adelian单值性决定的，相反，对于代数有限型的任何单值性，存在退化的黎曼曲面族，它实现了单值性。

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.