Automorphic function on the moduli space and analytic torsion

模空间上的自守函数和解析挠率

• 批准号: 10640071
• 负责人: YOSHIKAWA Kenichi
• 金额: $ 2.37万
• 依托单位: The University of Tokyo (1999)Nagoya University (1998)
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640071/
• 关键词: analytic torsion; automorphic form; moduli space; infinite dimensional algebra; infinite product; 一般Kac-Moody代数; Quillen計量; Borcherds Φ-関数; 2-elementary K3 曲面; Generalized Kac-Moody 代数; 複素双曲型空間

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 .

我们证明了一个事实，即具有抗对称性相关的K3表面（由固定曲线的校正），被视为模量空间上的函数，在IV型Hermitian类型领域的广泛含义中给出了自动形式。特别是，我们已经表明，对富函数的分析扭转与鲍尔切斯φ功能相吻合，并且该类似结果对各种2元素质晶格有效。结果，我们发现某些K3表面的分析扭转与Del Pezzo表面相关，与某些广义KAC-MOODY代数的分母函数相吻合。这些代数对应于带有Weyl载体的奇数单型双曲线晶格。连同一系列广泛的KAC-MOODY代数，甚至与Weyl Vector相对应，甚至与Weyl Vector相对应，我们的代数似乎形成了一系列有趣的广义KAC-MOODY代数，其字母功能由表征歧视界的自动形态形式给出。这些代数的几何含义远非理解。

项目成果

S. Kondo: "The automoylism prop of a generic Jacobian Kummn surfaces"J. Alg. Gem.. 7. 589-609 (1998)

S. Kondo：“通用雅可比库姆曲面的自酰主义支撑”J。

K.-I.Yoshikawa: "Smoothing of isolated hypersurface singularities and Quillen metrics"Asian Journal of Mathematics. 2. 325-344 (1998)

K.-I.Yoshikawa：“孤立超曲面奇点的平滑和 Quillen 度量”《亚洲数学杂志》。

S. Mukai: "Moduli Theory II"Iwanami Shoten Publ.. (2000)

S. Mukai：“Moduli Theory II”岩波书店出版..（2000）

K.-I.Yoshikawa: "Discriminant of theta divisors and Quillen metrics"Journal of Differential Geometry. (発売予定).

K.-I.Yoshikawa：“theta 除数和 Quillen 度量的判别式”《微分几何杂志》（待发布）。

向井 茂: "モジュライ理論 I"岩波書店. 174 (1998)

向井茂：《Modurai 理论 I》岩波书店 174 (1998)。