1999 Fiscal Year Final Research Report Summary
Study of singularities and geometry by means of representation theory
| Project/Area Number |
08404001
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | HOKKAIDO UNIVERSITY |
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Principal Investigator |
NAKAMURA Iku Grad. School of Sci., Hokkaido univ., Prof., 大学院・理学研究科, 教授 (50022687)
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| Co-Investigator(Kenkyū-buntansha) |
OKA Mutsuo Faculty of Sci., Tokyo Metro. Univ., Prof., 理学部, 教授 (40011697)
ISHIKAWA Goo Grad. School of Sci., Hokkaido univ., Assoc. Prof., 大学院・理学研究科, 助教授 (50176161)
SUWA Tatsuo Grad. School of Sci., Hokkaido univ., Prof., 大学院・理学研究科, 教授 (40109418)
KATSURA Toshiyuki Facult of Sci., Univ. of Tokyo, Prof., 大学院・数理科学研究科, 教授 (40108444)
EGUCHI Tohru Facult of Sci., Univ. of Tokyo, Prof., 理学部, 教授 (20151970)
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| Project Period (FY) |
1996 – 1999
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| Keywords | Representation / Singularity / Simple singularity / McKay correspondence / Abelian variety / moduli / Stability |
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| Research Abstract |
We studied two-dimensional McKay correspondence and a canonical compactification of the moduli of abelian varieties.
(I) For any finite subgroup of SL(2) we can construct a minimal resolution of a simple surface singularity CィイD12ィエD1/G as the Hilbert scheme of G-orbits. Via this construction we are able to provide a new explanation of two-dimensional McKay correspondence.
(ii) We proved that for any finite abelian subgroup G of SL(3) the Hilbert scheme of G-orbits is a crepant resolution of the singularity CィイD13ィエD1/G. For simple finite subgroup G of SL(3) (there are only two such) we determined the structure of the Hilbert scheme of G-orbits.
(iii) We constructed a canonical compactification SQィイD1g,NィエD1 of the moduli of abelian varieties over Z[ζィイD2NィエD2, 1/N]. Any point of SQィイD2g,NィエD2 is represented by an isomorphism class of a possibly singular abelian variety with certain level structure.
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