2001 Fiscal Year Final Research Report Summary
Study of moduli spaces, and K3 moonshine
| Project/Area Number |
10304001
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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 | KYOTO UNIVERCITY (2001)
Nagoya University (1998-2000) |
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Principal Investigator |
MUKAI Shigeru KYOTO UNIVERCITY, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (80115641)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAYAMA Noboru KYOTO UNIVERCITY, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (10189079)
HASHIMOTO Mitsuyasu Graduate School of Mathematics, Nagoya Univ., Associate Professor, 大学院・多元数理科学研究科, 助教授 (10208465)
KONDO Shigeyuki Graduate School of Mathematics, Nagoya Univ., Professor, 大学院・多元数理科学研究科, 教授 (50186847)
SAITO Masa-hiko Dept. of Math., Kobe Univ., Professor, 理学部, 教授 (80183044)
FUJINO Osamu KYOTO UNIVERCITY, Research Institute for Mathematical Sciences, Research Associate, 数理解析研究所, 助手 (60324711)
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| Project Period (FY) |
1998 – 2001
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| Keywords | moduli / vector bundle / invariant / Verlinde formula / conformal block / Hilbert's 14th problem / fundamental domain / K3 surface |
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| Research Abstract |
1. We reconstructed the geometric invariant theory and constructed the moduli space of vector bundles without using the Grothendieck's Quot-scheme. Both simplified the moduli theory of vector bundles a lot. We expect new development will be followed on this foundation. For example, it is interesting to study the degeneration of Jacobian using our description.
2. The construction of moduli spaces of vector bundles with additional structure, say parabolic structure or stable pair, were also simplified. By virtue of this, the celebrated Verlinde formula is now regarded as the Cayley-Sylvester type explicit formula for a certain invariant ring. We hope that many mathematics around the formula, including the affine Lie algebra, Hecke algebra and quantum group, will become theorems in a modern invariant theory.
3. The master space of the moduli of rank two parabolic vector bundles over punctured Riemann sphere, or equivalently pointed projective line, exists. Its coordinate ring is the invaria
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nt ring of a certain square zero linear action of the 2-dimensional additive group on a polynomial ring. In particular, the invariant ring is finitely generated. Together with the results mentioned below, we have solved the (original) Hilbert fourteenth problem for the square free action of multi-dimensional additive groups.
4. We constructed a counterexample of Hilbert's fourteenth problem for the 3-dimensional additive group. This ring is isomorphic to the total coordinate ring of the blow-up of the 5-dimensional projective space at nine points. We also gave a simplified proof of this isomorphism.
5. We found a new proof of the Shafarevich conjecture on the algebraicity of a certain class of Hodge cycles on the product of two K3 surfaces.
6. We defined a bi-level structure of an abelian variety and studied the moduli of abelian surfaces equipped with this structures. The moduli spacce is very simple and has a lot of geometry when the polarization type is (1,d) and d 【less than or equal】 5. It is very interesting to apply the trace formula to this moduli problem and determine the multiplicative structure of the ring of automorphic forms. Less
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