Research on Characteristic Classes of Singular Varieties

单一品种特征类研究

• 批准号: 11440014
• 负责人: SUWA Tatsuo
• 金额: $ 7.55万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440014/
• 关键词: singular varieties; localization of characteristic classes; Schwartz-MacPherson class; Fulton-Johnson class; Milnor class; multiplicity; coherent sheaves; Riemann-Roch theorem; 特性類の局所化; Fulton-Johnson類; Milnor数; Riemanu-Rochの定理

Directed by the head investigator Suwa, the research on characteristic of singular varieties, in particular the theory of Milnor classes and related topics have been performed, as described in the research proposal, We obtained an explicit formula for the Minor class of a non- singular component of singular variety, the notion of the homology Chern class is introduced. The Riemann-Roch theorem for embeddings of singular varieties are proved arid used to compute the Chern class of the tangent sheaf of a singular variety. As an application of the theory of localization of characteristic classes, we defined, for functions on singular varieties the notion of multiplicity at the singularity, gave the method to compute them and proved, in the global situation, the "multiplicity formula", which generalized well-known classical formula in the non-singular case. We also gave a direct and geometric proof of the Lefschetz fixed point formula for the de Rham and Dolbeault complexes.The other investigators collaborated in the above projects and also obtained many other results in the subjects such as : the moduli space of Abelian varieties, me McKay correspondence of simple singularities, developable surfaces of curves in the Euclidean space and classification of singularities of Dalboux sphere representations, stability of singular Lagrangian varieties, singular fibers of elliptic K3 surges and the torsion subgroup of the Mordell-Weli group, geometry of sextic curves of torus type and the geometry of dual curves, an alternative proof of the algebraic independence of the Monta-Mumfoixl classes, parital answer to the Akita conjecture on the mapping class group.

在Suwa教授的指导下，对奇异簇的特征，特别是Milnor类的理论及相关问题进行了研究，得到了奇异簇的非奇异分支的Minor类的一个显式公式，引入了同调Chern类的概念.证明了奇异簇嵌入的Riemann-Roch定理，并利用该定理计算了奇异簇切层的Chern类。作为特征类局部化理论的一个应用，我们对奇异簇上的函数定义了奇异点处的重数概念，给出了计算方法，并在整体情形下证明了“重数公式”，它推广了非奇异情形下著名的经典公式.我们还给出了de Rham复形和Dolbeault复形的Lefschetz不动点公式的一个直接的几何证明。其他研究人员在上述项目中合作，也获得了许多其他结果，如：阿贝尔簇的模空间，简单奇点的McKay对应，欧氏空间中曲线的可展曲面和Dalboux球表示的奇点分类，奇异Lagrange簇的稳定性，椭圆K ~ 3波的奇异纤维和Mordell-Weli群的挠子群，环面型六次曲线的几何和对偶曲线的几何，Monta-Mumfoixl类代数独立性的另一种证明，映射类群上秋田猜想的部分回答。

项目成果

T.Suwa: "Characteristic classes of coherent sheaves on singular varieties"Singularities-Sapporo 1998, ASPM. 29. 279-297 (2000)

T.Suwa：“奇异品种上相干滑轮的特征类别”Singularities-Sapporo 1998，ASPM。

G.Ishikawa: "Topological classification of the taugent developables of space curves"J.London Math.Soc.. 62. 583-598 (2000)

G.Ishikawa：“空间曲线的 taugent 可展性的拓扑分类”J.London Math.Soc.. 62. 583-598 (2000)

I.Nakamura: "Hilbert scheme of G-orbits in dimension three"Asian J.Math. 4. 51-70 (2000)

I.Nakamura：“第三维 G 轨道的希尔伯特方案”亚洲 J.Math。

M. Oka: "Flex curves and their applications"Geometriae Dedicata. 75. 67-100 (1999)

M. Oka：“弯曲曲线及其应用”Geometriae Dedicata。

S.Yokura: "On a Verdier-type Riemann-Roch for Chern-Schwartz-MacPherson class"Topology and Its Applications. 94. 315-327 (1999)

S.Yokura：“关于 Chern-Schwartz-MacPherson 类的 Verdier 型 Riemann-Roch”拓扑及其应用。