2003 Fiscal Year Final Research Report Summary
A STUDY ON THE GEOMETRY OF MODULI SPACES
| Project/Area Number |
12304001
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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 |
Algebra
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| Research Institution | HOKKAIDO UNIVERSITY |
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Principal Investigator |
NAKAMURA Iku Hokkaido Univ., Grad.School of Sci., Prof., 大学院・理学研究科, 教授 (50022687)
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| Co-Investigator(Kenkyū-buntansha) |
KATSURA Toshiyuki Tokyo Uiv., Grad.School of Math.Sci., Prof., 大学院・数理科学研究科, 教授 (40108444)
SHINODA Ken-ichi Sophia Univ., Fac.of Sci and Tech., Prof., 理工学部, 教授 (20053712)
SUWA Tatsuo Hokkaido Univ., Grad.School of Sci., Prof., 大学院・理学研究科, 教授 (40109418)
NAKAJIMA Hiraku Kyoto Univ., Grad.School of Sci., Prof., 大学院・理学研究科, 教授 (00201666)
SAITO Masahiko Kobe Univ., Fac.of Sci., Prof., 理学部, 教授 (80183044)
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| Project Period (FY) |
2000 – 2003
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| Keywords | Abelian variety / Moduli / Compactification / McKay correspondence / Theta function / Calabi-Yau Manifolds / Coinvariant algebra / Quiver variety |
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| Research Abstract |
Certain compactification of moduli space of abelian varieties was studied as well as moduli spaces of G-orbits for a finite subgroup G of SL(2,C) and SL(3,C). The main issues we have in mind are as follows (a) Study of a resolution of singularity of the quotient C^3/G as a moduli space (b) study of Kempf stability and compactification of moduli spaces (c) A canonical ompactification SQ_<g,N> of the moduli A_<g,N> over Z[1/N] of abelian varieties and related moduli.
There were remarkable progresses on each subject during this project. The main results are as follows : first there was a remarkable progress in the study on Hilbert schemes of G-orbits. We copuld give a new explanation to the phenomenon of McKay correspondence which was discovered over twenty years, and extending it to the three dimensional case, we obtained a lot of new resluts. The head investigator (Nakamura) proposed a generalization of McKay correspondence to the three or higher dimension, which was follows by many related results. In this sense this project payed a substantial role in the history of studying McKay correspondence. Among other things Nakamura showed that the Hilbert scheme of G-orbits is the canonical resolution of singularities of the quotient C^3/G. This is a new discovery which has never been observed, against the common sense in minimal model theory. Therefore this discovery has been accepted by specialists with surprise. Another substantial contribution of this project is that we constructed a new canonical compactification of moduli space A_<g,N> of abelian varieties This compactification is projective, it enjoys a desirable property as a compactification. From the stabdpoint of invariant theory, this compactification is ust that by stability. In this sense it is orthodox and is uniquely characterized by this property
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