Geometric Study of Algebraic Systems

1.首席研究员Masayoshi Miyanishi对基于开代数曲面分类理论的Abhyankar-Moh定理和Lin-Zauberberg定理给出了新的证明。他认为嵌入的仿射曲线与一个地方点在无穷远到仿射平面和分类等嵌入可能最低程度时，属低。此外，他还简化了P.Roberts对Hilbert第十四问题的反例的证明。2.藤木彰发现了超Kahler流形作为四元数流形的一个自然的部分紧化，这些流形是作为某些类型的超Kahler quotinet空间得到的。他还利用等变上同调群研究了Kahler商空间通过矩映射的变化。作为这一结果的应用，他阐明了Hodge结构的变分的Z-结构和单值性的描述。4. Jun Murakami基于纽结的Kontsevich不变量构造了一个新的三维流形不变量，并研究了它的性质。他利用这一不变量建立了拓扑量子理论，并将其应用于曲面的映射群族。5. Kazuhiro Konno对非超椭圆束曲面定义了一个新的指数，并找到了研究曲面斜率下界的一般方法。他还指数，称为堀川指数，退化纤维。6. Hiroyuki Yamane获得了一个新的技术来计算指数的宝马代数中定义的一份文件中的拉姆。

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.