Residues on Singular Varieties

单一品种的残留

• 批准号: 15340016
• 负责人: SUWA Tatsuo
• 金额: $ 5.18万
• 依托单位: Niigata University (2005)Hokkaido University (2003-2004)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340016/
• 关键词: Singular varieties; Localization of characteristic classes; Residues; Chern classes; Local Euler obstructions; Holomorphic foliations; Complex dynamical systems; Intersection theory; 交点理論; 切断の組; Thom-Porteous公式

The head investigator and the others did research on residues on singular varieties. More specifically :1.We developed a theory of residues of Chern classes of vector bundles on singular varieties with respect to collections of sections. We also gave explicit expressions (analytic, algebraic and topological) of the residues at an isolated singular point.2.We introduced the notion of multiplicity for functions on singular varieties and proved a generalization of a formula of Iversen for holomorphic maps onto a Riemann surface.3.In the case of the top Chern class, the Thom class contains local informations and produces analytic., algebraic and topological invariants through the Bochner-Martinelli kernel. We found the "intermediate Thom classes" for other Chern classes.4.In a collaboration with J.-P.Brasselet and J.Seade, we proved a "proportionality theorem" for the local Euler obstruction of 1-forms on singular varieties.5.In a collaboration with F.Bracci, we prove the existence of parabolic curves at a fixed point of a holomorphic self-map of a singular complex surface, as an application of our residue theory. For this, we developed the intersection theory of curves in singular surfaces, using the Grothendieck residues on singular varieties.6.Besides the above, Ito obtained important results on the Poincare-Hopf type theorems, Ohmoto on the characteristic classes of algebraic stacks, Oka on the fundamental group of the complement of algebraic curves, Tajima on Milnor and Tjurina numbers, Yokuraon motivic characteristic classes, respectively.

首席调查员和其他人对单一品种上的残留进行了研究。更具体地说：1.我们发展了奇异变种上的向量丛的陈氏类关于截集的剩余理论。2.我们引入了奇异簇上函数的重数的概念，证明了黎曼曲面上全纯映射的Iversen公式的推广.3.对于top Chern类，Thom类包含局部信息，并通过Bochner-Martinelli核产生解析、代数和拓扑不变量.4.与J.-P.Brasselet和J.Seade合作，证明了奇异簇上1-形式的局部Euler阻塞的一个“比例定理”.5.与F.Bracci合作，证明了奇异复曲面的全纯自映射的不动点上抛物曲线的存在性，作为我们剩余理论的一个应用.为此，我们利用奇异变量上的Grothendieck留数发展了奇异曲面上曲线的交理论。6.除此之外，Ito在Poincare-Hopf型定理、Ohmoto关于代数堆栈的特征类、Oka关于代数曲线补的基本群、Tajima关于Milnor和Tjuina数、Yokuraon动机特征类等方面分别得到了重要的结果。

项目成果

Classifying singular Legendre curves by contactomorphisms

通过接触同态对奇异勒让德曲线进行分类

• 发表时间: 2004
• 期刊: Journal of Geometry and Physics 52
• 作者: Y. Ebihara;Y. Miyoshi;K. Asamura;et al;S.Severmann et al.;G.Ishikawa
• 通讯作者: G.Ishikawa

T.Suwa: "Characteristic classes of singular varieties"Sugaku Expositions, A.M.S.. 16. 153-175 (2003)

T.Suwa：“奇异品种的特征类别”Sugaku Expositions，A.M.S.. 16. 153-175 (2003)

A Poincare-Hopf type theorem for holomorphic one-forms

全纯一型的Poincare-Hopf型定理

• 发表时间: 2005
• 期刊: Topology 44
• 作者: T.Ito;B.Scardua
• 通讯作者: B.Scardua

Supersingular K3 surfaces in odd characteristic and sextic double plane

奇特征和六重双平面中的超奇异 K3 表面

• 发表时间: 2004
• 期刊: Math.Ann. 328
• 作者: 大本 亨;T.Ohmoto;諏訪 立雄;伊藤 敏和;岡 睦雄;田島 慎一;與倉 昭治;T.Suwa;T.Ito;M.Oka;S.Tajima;S.Yokura;T.Suwa;I.Nakamura;G.Ishikawa;G.Ishikawa;I.Shimada;I.Shimada
• 通讯作者: I.Shimada

I.Shimada: "Fundamental groups of algebraic fiber spaces"Comment.Math.Helu.. 78. 335-362 (2003)

I.Shimada：“代数纤维空间的基本群”评论.Math.Helu.. 78. 335-362 (2003)