Problems surrounding manifolds

在两年的合作研究中，我们在以下主题上取得了稳定而显著的进展。所提出的三维取消问题被认为是与仿射空间上的代数环面作用。在这个过程中，我们考虑了同调平面和可收缩的代数曲面，主要研究者成功地将其分类在科代拉维小于2的情况下。在一般类型的情况下，我们发现了一个无限系列的例子，这样的表面。2.在一般型代数曲面的半稳定退化的研究中，Tsunoda和Zhang得到了Noether不等式的一个对数模拟，利用它可以在非完全情况下发展一个理论，与Horikawa曲面的理论相对应.正亏格的穿孔Riemann曲面的基本群为构造有理数域的Galois扩张提供了重要的信息。Kaneko利用子群的过滤研究了基本群的1-adic完备化，并阐明了子商群的结构。4 & 5.川中工作的Hecke代数的一般线性群定义在有限领域。Kawakubo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，获得了关于3维或4维流形的新结果.具有Einstein-Kaehler度量的流形近年来引起了人们的广泛关注。坂根是第一个在构建非齐次爱因斯坦-凯勒流形的例子。7 & 8。我们在流形上的整体分析理论方面取得了显著的进展。尤其是植木给出了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.