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Geometric Study of Algebraic Systems

代数系统的几何研究

基本信息

批准号：
07454006
负责人：
MIYANISHI Masayoshi
金额：
$ 3.97万
依托单位：
Osaka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1995
资助国家：
日本
起止时间：
1995 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-07454006/
关键词：
Theorem of Abhyankar-Moh Open algebraic surfaces Hyper Kahler manifold Log geometry Hodge structure Quantum invariant Surface of general type BMW algebra アフィン空間代数群の作用射影代数多様体

项目摘要

1.The head investigator, Masayoshi Miyanishi, gave new proofs to the Abhyankar-Moh Theorem and the Lin-Zaidenberg Theorem which are based on the classification theory of open algebraic surfaces. He considered the embeddings of affine curves with one-place points at infinity into the affine plane and classified such embeddings of possibly minimal degree when the genus is low. He, furthermore, simplified a proof in P.Roberts' counterexample to the fourteenth problem of Hilbert.2.Akira Fujiki found a natural partial compactification of hyper kahler manifolds as quaternionic manifolds which are obtained as hyper Kahler quotinet spaces of certain kinds. He also investigated the variation via moment maps of Kahler quotient spaces by making use of equivariant cohomology groups.3.Sanpei Usui proved that one can recover vanishing cycles by using log geometry. As applications of this result, he clarified the Z-structure of the variation of Hodge structures and the description of monodromies.4.Jun Murakami constructed a new invariant for three-dimensional manifolds which is based on the Kontsevich invariant for the knots and studied its properties. He constructed a topological quantum theory by making use of this invariant, and a family of mapping groups of surfaces as its application.5.Kazuhiro Konno defined a new index for surfaces with non-hyperelliptic pencils and found some general method to investigate the lower bound of the slopes of surfaces. He also an index, called the Horikawa index, for degenerate fibers.6.Hiroyuki Yamane obtained a new technique to calculate the index of the BMW algebras which are defined in a paper of Rham.

1.首席研究员Masayoshi Miyanishi在开放代数曲面分类理论的基础上，对Abhyankar-Moh定理和Lin-Zaidenberg定理进行了新的证明。他考虑了无穷远处有单点的仿射曲线在仿射平面上的嵌入，并对这类嵌入在属低时可能最小度的嵌入进行了分类。他进一步简化了P.Roberts对hilbert第十四问题的反例中的一个证明。藤木明（Akira Fujiki）发现了超kahler流形作为四元数流形的自然偏紧化，这种四元数流形是由某些类型的超kahler引用空间得到的。他还利用等变上同调群研究了Kahler商空间的矩映射的变异。臼井三培证明了利用对数几何可以恢复消失的循环。作为这一结果的应用，他阐明了Hodge结构变异的z型结构和单态的描述。Jun Murakami在结的Kontsevich不变量的基础上构造了三维流形的不变量，并研究了它的性质。他利用这一不变量构造了拓扑量子理论，并将一系列曲面映射群作为其应用。Kazuhiro Konno为带有非超椭圆铅笔的曲面定义了一个新的指标，并找到了一些研究曲面斜率下界的一般方法。他还为简并纤维提出了一个指数，称为堀川指数。Hiroyuki Yamane在Rham的一篇论文中获得了一种计算BMW代数指标的新方法。