Problems surrounding manifolds

在两年的合作研究中，我们在以下主题上取得了稳步、值得注意的进展。所提出的三维抵消问题是与仿射空间上的代数环面作用有关的。在这个过程中，我们考虑了同调平面和可压缩代数曲面，主要研究者在Kodaira维数小于2的情况下成功地对它们进行了分类。在一般类型的情况下，我们找到了这类曲面的无限系列例子。在一般类型代数曲面的半稳定退化的研究中，Tsunoda和Zhang得到了Noether不等式的对数模拟，利用它可以发展出与Horikawa曲面的非完备情形相对应的理论。正亏格的穿孔黎曼曲面的基本群将为构造有理数域的Galois扩张提供重要的信息。Kaneko利用子群的滤子研究了基本群的1-Add完备性，并阐明了子群的结构。4&5.Kawanaka研究了定义在有限域上的一般线性群的Hecke代数。川久保在流形上李群作用的拓扑研究方面取得了新的进展。小林通过对纽结的研究，得到了3维或4维流形上的新结果。最近，具有爱因斯坦-凯勒度量的流形引起了人们的极大关注。Sakane是第一个构造非齐次Einstein-Kaehler流形的例子的人。7&8。我们在流形上的整体分析理论方面取得了显着的进展。特别是，Ueki给出了Riemann-Roch定理和Gauss-Bonnet定理等大定理的概率证明。

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.