Analytic torsion and discriminant

2021 Fiscal Year Final Research Report

• Project/Area Number: 16H03935
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyoto University
• Principal Investigator: Yoshikawa Ken-Ichi 京都大学, 理学研究科, 教授 (20242810)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: 解析的捩率 / BCOV不変量 / 判別式 / 保型形式 / Borcherds積 / Calabi-Yau多様体 / Enriques多様体 / 対数的Enriques曲面

Outline of Final Research Achievements

We constructed a holomorphic torsion invariant for log-Enriques surfaces and proved that this invariant is expressed as the Petersson norm of an explicit Borcherds product on the Kaehler moduli space of the corresponding Del Pezzo surface. We constructed a holomorphic torsion invariant for Enriques manifolds of higher dimension and proved that this invariant is a potential function of the Weil-Petersson metric on their moduli space. For some non-Borcea-Voisin Calabi-Yau orbifolds of dimension three, we computed the BCOV invariant. We proved that the quasi-pullback of the product of the even theta constants via the Torelli map for 2-elementary K3 surfaces is given by the product of even theta constants of smaller genus and a Borcherds product. We gave a factorization formula for the difference of j-invariants in terms of the pullback of the Borcherds Phi-function to the product of complex upper half plane.

Free Research Field

複素幾何学

Academic Significance and Societal Importance of the Research Achievements

対合付K3曲面と3次元Calabi-Yau多様体に限られていた解析的捩率不変量の構成法をあるクラスの特異Calabi-Yau空間や高次元Enriques多様体に拡張する事で、より広範なクラスの多様体に対して解析的捩率不変量が存在し、モジュライ空間上に興味深い関数が存在する事が示された。これまでIV型領域上の保型形式に限られていた無限積展開を持つ保型形式の理論をIV型領域上の別の直線束の切断に拡張できる可能性が示唆された。