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

算术和拓扑中各种基本群及其结构的比较研究

基本信息

批准号：
01460002
负责人：
KAWAMATA Yujiro
金额：
$ 4.42万
依托单位：
University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (B)
财政年份：
1989
资助国家：
日本
起止时间：
1989 至 1990
项目状态：
已结题

项目摘要

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.

在维度 3 的情况下，最小模型具有奇异点，每个奇异点都有称为索引的不变量。尽管在非奇异模型中消失，但已知整个指数 = 篮子是双有理不变量，也是变形不变量。顺便说一句，在 3 维簇的分类中，需要某些类簇的指数有界性。我之前已经证明，当规范除数 k 在数值上等于零时，索引是有界的。作为这一点的延续，我在论文“Q-Fano Threefolds 的有界性”中表明，当 K 为负时，索引和 K^3 都是有界的。我在《代数曲面的适度退化》一文中也用完全不同的方法证明了椭圆曲面退化的指数有界性。此外，在同一篇论文中，我已经证明表面退化的最小模型的拓扑与半稳定退化的拓扑非常相似。如果 X 是在代数数域 k 上定义的代数簇，则基本群 pi, ( X)可以被认为是绝对伽罗瓦群Gal (k/k)的群扩展。 Nakamura 研究了 X 可以用 pi (X) 的群论性质来表征的条件，并获得了一些结果。首先，他证明当X是某条双曲曲线时，pi,(X)作为Gal(k/k)的群外延，唯一确定X直至同构。他还证明，当X是P'减三点时，作为群外延的pi,(X)的每一个自同构都必须从X本身的自同构中导出。松本研究了黎曼曲面简并族中奇异函数的拓扑分类；他证明了奇异五元的拓扑类型是由非阿迪利亚一元性决定的，反之，对于任何代数有限类型的一元性，都存在一个实现该一元性的黎曼曲面的简并族。