2007 Fiscal Year Final Research Report Summary
Residues on Singular Varieties
| Project/Area Number |
18340015
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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 |
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Principal Investigator |
SUWA Tatsuo Niigata University, Institute of Science and Technology, Professor (40109418)
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| Co-Investigator(Kenkyū-buntansha) |
OHMOTO Toru Hokkaido University, Graduate School of Science, Assoc. Professor (20264400)
OKA Mutsuo Tokyo University of Science, Faculty of Science, Professor (40011697)
SAITO Kyoji Kyoto University, RIMS, Professor (20012445)
TAJIMA Shinichi Niigata University, Institute of Science and Technology, Professor (70155076)
YOKURA Shoji Kagoshima University, Faculty of Science, Professor (60182680)
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| Project Period (FY) |
2006 – 2007
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| Keywords | singular varieties / localization of characteristic classes / residues / Chern classes / Atiyah classes / 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 and related subjects. More specifically:
1. We developed a theory of localization of Chern classes by frames of vector bundles on singular varieties. In the previous year, we gave explicit expressions (analytic, algebraic and topological)of the residues at an isolated singularity. In this research, we gave an expression in the case the singularity is not isolated.
2. As an application of 1 above, we developed an analytic intersection theory on singular varieties. This clarifies global intersections, local intersections and the relation between the two. In the global case, the localization theory of Chern classes is very effective and in the local case, Grothendieck residues on singular varieties play an essential role. The two situations are related by the residue theorem.
3. As a summary of collaboration with J.-P. Brasselet and J. Seade, we almost finished writing a book on the characteristic classes of singular varieties utilizing indices and residues of vector fields. This also includes a new simple proof of the Proportionality Theorem, which describes a fundamental property of the local Euler obstruction of singular varieties.
4. In the collaboration with M. Abate, F. Bracci and F. Tovena, we started to construct a localization theory of Atiyah classes of holomorphic vector bundles. This theory is expected to be very interesting and have many applications.
5. Besides the above, Ohmoto obtained important results on the characteristic classes of varieties with group actions, Oka on the fundamental group of the complement of algebraic curves, Saito on Lie algebras and singularities, Tajima on Milnor and Tjurina numbers, Yokura on motivic characteristic classes, respectively.
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