Problems surrounding manifolds

在两年的合作研究中，我们在以下课题上取得了稳定、显著的进展。考虑了仿射空间上的代数环面作用下的三维消去问题。在课程中，我们考虑了同调平面和可压缩代数曲面，在Kodaira维数小于2的情况下，主要研究者成功地对它们进行了分类。在一般类型的情况下，我们找到了无数个这样的曲面的例子。在对一般代数曲面的半稳定退化的研究中，Tsunoda和Zhang得到了Noether不等式的一个对数模拟，利用它可以在非完全情况下建立一个与Horikawa曲面相对应的理论。穿孔的正属黎曼曲面的基群将为构造有理数域的伽罗瓦扩展提供重要的信息。Kaneko利用子群过滤法研究了基本群的1进补全，阐明了子商群的结构。4 & 5。Kawanaka研究了有限域上定义的一般线性群的Hecke代数。川久保玲在流形上李群作用的拓扑研究上取得了新的进展。小林通过对结的研究，获得了关于3维或4维流形的新结果。具有爱因斯坦-凯勒度量的流形最近引起了人们的广泛关注。Sakane是第一个构造非齐次Einstein-Kaehler流形的人。7 & 8。我们在流形的全局分析理论方面取得了显著的进展。特别地，Ueki给出了像黎曼-洛克定理和高斯-邦尼特定理这样的大定理的概率证明。

During two years of cooperative research, we had steady, noteworthy progress in the following topics.1. The proposed 3-dimensional cancellation problem was considered in connection with algebraic torus action on the affine space. In the course, we considered homology planes and contractible algebraic surfaces, which the principal investigator succeeded in classifying in the case of Kodaira dimension less than 2. In the case of general type, we found an infinite series of examples of such surfaces.2. In a study of semi-stable degenerations of algebraic surfaces of general type, Tsunoda and Zhang obtained a logarithmic analogue of the Noether's inequality, with which one is able to develop a theory in the non-complete case corresponding to the one of Horikawa surfaces.3. The fundamental group of a punctured Riemann surface of positive genus will provide important informations for constructing the Galois extensions of the rational number field. Kaneko made a study on the 1-adic completion of the fundamental group by making use of a filtration by subgroups and clarified the structure of subquotient groups.4 & 5. Kawanaka worked on the Hecke algebra of a general linear group defined over a finite field. Kawakubo made a new progress in the topological research of Lie group actions on manifolds. Kobayashi obtained new results on manifolds of dimension 3 or 4 through his study on knots.6. Manifolds with Einstein-Kaehler metrics attract recently lots of attention. Sakane was the first in constructing examples of non-homogeneous Einstein-Kaehler manifolds.7 & 8. We have made remarkable progress in the theory of global analysis on manifolds. In particular, Ueki gave probabilistic proofs of the big theorems like Riemann-Roch theorem and Gauss-Bonnet theorem.