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.

首席研究员和其他人对单一品种的残留物进行了研究。1.建立了向量丛的Chern类在奇异簇上关于截集的剩余理论。给出了孤立奇点上剩余的解析、代数和拓扑表达式. 2.引入了奇异簇上函数重数的概念，并证明了Riemann曲面上全纯映射的Iversen公式的推广. 3.在顶Chern类中，Thom类包含局部信息，并产生解析.通过Bochner-Martinelli内核的代数和拓扑不变量。我们发现了其他陈省身类的“中间Thom类”。P. Atelet和J.Seade，我们证明了奇异变元上1-形式的局部Euler阻塞的“比例定理”。5.与F.Bracci合作，我们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，作为我们的留数理论的应用。6.除此之外，Ito在Poincare-Hopf型定理、Ohmoto在代数栈的特征类、Oka在代数曲线的补的基本群、Tajima在Milnor数和Tjurina数、Yokuraon motivic特征类等方面分别得到了重要结果.

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)

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)

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