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.

在尺寸3的情况下，最小模型具有奇异的点，每个单数点具有不变的索引。尽管在非单明模型中消失了，但众所周知，整个索引=篮子是birational不变的，也是变形不变的。顺便说一句，在三维品种的CALSSIFIENS中，需要某些类别的索引的界限。我之前已经表明，当规范除数k在数值上等于零时，索引是界定的。为此，我已经证明，当k在“ q-fano三倍的界限”中k中为负时，索引和k^3都是有限的。通过完全不同的方法，我还显示了椭圆表面退化的指标的界限，“代数表面的中等变性”。此外，在同一篇论文中，我已经表明，表面变性的最小模型的拓扑结构与可半合并的变性非常相似。如果X是代数数字k上定义的代数品种，则可以将基本组PI（x）视为绝对Galois Glopal Galois Galois Gal（K/K）的基本组PI（X）。中村研究了x可以通过pi（x）的组理论特性来表征x的条件，并获得了几个结果。首先，他表明，当x是一定的双曲线曲线时，pi，（x）作为gal（k/k）的组延伸，将x确定为唯一的同构。他还表明，当x是p'minus三分时，必须从x本身的自动晶体中诱导pi的每一个自动形态（x）。他证明了奇异型膜的拓扑类型是由非阿德式单曲率确定的，相反，对于任何代数有限类型的单片，存在一个脱发的riemann表面家族，可以实现单型。