2005 Fiscal Year Final Research Report Summary
Residues on Singular Varieties
| Project/Area Number |
15340016
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Niigata University (2005)
Hokkaido University (2003-2004) |
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Principal Investigator |
SUWA Tatsuo Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (40109418)
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| Co-Investigator(Kenkyū-buntansha) |
ITO Toshikazu Ryukoku University, Faculty of Economy, Professor, 経済学部, 教授 (60110178)
OHMOTO Toru Hokkaido University, Graduate School of Science, Assoc.Professor, 大学院・理学研究科, 助教授 (20264400)
OKA Mutsuo Tokyo University of Science, Faculty of Science, Professor, 理学部, 教授 (40011697)
TAJIMA Shinichi Niigata University, Institute of Science and Technology, Professor, 自然科学系, 教授 (70155076)
YOKURA Shoji Kagoshima University, Faculty of Science, Professor, 理学部, 教授 (60182680)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Singular varieties / Localization of characteristic classes / Residues / Chern classes / Local Euler obstructions / Holomorphic foliations / Complex dynamical systems / Intersection theory |
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| Research Abstract |
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, Yokura
on motivic characteristic classes, respectively.
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