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Analytic Torsion and Automorphic Forms with Infinite Products

具有无穷积的解析扭转和自守形式

基本信息

批准号：
12640061
负责人：
YOSHIKAWA Ken-ichi
金额：
$ 2.3万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640061/
关键词：
Analytic Torsion Quillen metric Moduli Space Automorphic Form Borcherds Product K3 Surface

项目摘要

(1) In 1998, we introduced an invariant of a K3 surface with anti-symplectic involution : By fixing a Ricci-flat Kaehler metric on a K3 surface, which is invariant under the involution, the notion of the equivariant analytic torsion of the K3 surface with involution and of the analytic torsion of the fixed curves make sense. Then, the product of these two quantities is our invariant. In these two years, it has become possible to define the invariant without using Yau's theorem, the existence of Ricci-flat Kaehier metrics on a K3 surface. Indeed, by adding certain factor of Bott-Chern class to the previous definition, onecan obtain the same invariant without assuming the Ricci-flatness of the metric. This invariant is represented by an automorphic form on the moduli space. Before this progress, it was inevitable to study the degenerating behavior of Ricci-flat metrics, which made our proof hard to read. The fact that the invariant is independent of the choice of a metric, reduces the study of its degenerating behavior to that of Bott-Chern classes. It is much easier to understand the degenerations of Bott-Chern classes than that of Einstein metrics.(2) It was known before that the analytic torsion of curves of genus 1 (resp. 2) is represented by a certain Siegel modular form. By a jointwork with Shu KAWAGUCHI (Kyoto Univ.), we extend this fact to curves of genus 3. More precisely, their Quillen metric is represented by a certain Siegel modular form. The key fact is that every non-hyperelliptic curve of genus 3 is a hyperplane section of a Kummer's quartic. However, that realization depends on the choice of an unramified double covering of the curve.

(1)在1998年，我们引入了一个反辛对合K3曲面的不变量：通过在一个对合下不变的K3曲面上固定一个Ricci平坦的Kaehler度量，对合K3曲面的等变解析挠率和固定曲线的解析挠率的概念变得有意义。那么，这两个量的乘积就是我们的不变量。在这两年中，它已经成为可能，以确定不变量，而不使用丘的定理，存在的Ricci平坦Kaeglund度量的K3表面。实际上，在上述定义中加入Bott-Chern类的因子，就可以得到相同的不变量，而不需要假设度量的Ricci平坦性。这个不变量由模空间上的自守形式表示。在这一进展之前，研究Ricci平坦度量的退化行为是不可避免的，这使得我们的证明难以阅读。事实上，不变量是独立的度量的选择，减少了研究其退化行为的Bott-Chern类。Bott-Chern类的退化比Einstein度量的退化更容易理解。(2)已知亏格为1的曲线的解析挠率（分别为2)是由某种西格尔模形式表示的。通过与Shu川口（京都大学）的合作，我们将这一事实推广到亏格为3的曲线。更准确地说，他们的奎伦度量由某种西格尔模形式表示。关键的事实是，每个亏格为3的非超椭圆曲线是库默四次曲线的超平面截面。然而，这种实现取决于曲线的非分歧双重覆盖的选择。