1999 Fiscal Year Final Research Report Summary
Automorphic function on the moduli space and analytic torsion
| Project/Area Number |
10640071
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | The University of Tokyo (1999)
Nagoya University (1998) |
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Principal Investigator |
YOSHIKAWA Kenichi Graduate School of Math. Sci., The University of Tokyo, Associate Prof., 大学院・数理科学研究科, 助教授 (20242810)
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| Co-Investigator(Kenkyū-buntansha) |
NAMIKAWA Yukihiko Graduate School of Math., Nagoya Univ., Prof., 大学院・多元数理科学研究科, 教授 (20022676)
KOBAYASHI Ryoichi Graduate School of Math., Nagoya Univ., Prof., 大学院・多元数理科学研究科, 教授 (20162034)
OHSAWA Takeo Graduate School of Math., Nagoya Univ., Prof., 大学院・多元数理科学研究科, 教授 (30115802)
KONDO Shigeyuki Graduate School of Math., Nagoya Univ., Associate Prof., 大学院・多元数理科学研究科, 助教授 (50186847)
MUKAI Shigeru Graduate School of Math., Nagoya Univ., Prof., 大学院・多元数理科学研究科, 教授 (80115641)
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| Project Period (FY) |
1998 – 1999
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| Keywords | analytic torsion / automorphic form / moduli space / infinite dimensional algebra / infinite product |
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| Research Abstract |
We have proved the fact that analytic torsion of a K3 surface with anti-symplectic involution (corrected by that of the fixed curve), regarded as a function on the moduli space, is given by an automorphic form in the wider sense on the Hermitian domain of type IV . In particular, we have shown that analytic torsion of an Enriques surface coincides with Borcherds Φ-function, and that analogous result is valid for various 2-elementary lattices. As a result, we found that analytic torsion of certain K3 surfaces related, with Del Pezzo surfaces coincides with the denominator function of some generalized Kac-Moody algebras . These algebras correspond to the odd unimodular hyperbolic lattices with Weyl vector. Together with a series of generalized Kac-Moody algebras corresponding to even unimodular hyper-bolic lattices with Weyl vector, our algebras seem to form an interesting series of generalized Kac-Moody algebras whose denominator function is given by the automorphic form characterizing the discriminant locus. The geometric meaning of these algebras is far from understanding .
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