Analytic torsion and discriminant

• 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
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥9,100,000 (Direct Cost: ¥7,000,000、Indirect Cost: ¥2,100,000); Fiscal Year 2020: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2019: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2018: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• 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 / テータ定数 / モジュライ空間 / エンリケス多様体 / 対数的エンリケス曲面 / ボルチャーズ積 / クンマー曲面 / エンリケス曲面 / j-不変量

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.

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。

Research Products

• [Int'l Joint Research] カリフォルニア大学サンタバーバラ校(米国)
• [Int'l Joint Research] Rennes University/CNRS(フランス)
• [Int'l Joint Research] Chalmers University of technology(スウェーデン)
• [Int'l Joint Research] カリフォルニア大学/サンタバーバラ校(米国)
• [Journal Article] K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: the structure of invariant (2020)
  • Author(s): Souhei Ma, Ken-Ichi Yoshikawa
  • Journal Title: Compositio Mathematica Volume: 156 Issue: 10 Pages: 1965-2019
  • DOI: 10.1112/s0010437x2000737x
  • NAID: 120007193026
  • Peer Reviewed
• [Journal Article] K3 surfaces with involution and analytic torsion (2020)
  • Author(s): Ken-Ichi Yoshikawa
  • Journal Title: Sugaku Expositions Volume: 33 Issue: 1 Pages: 85-109
  • DOI: 10.1090/suga/449
  • NAID: 130007420317
  • Peer Reviewed
• [Journal Article] Resultants and the Borcherds Φ-function (2018)
  • Author(s): Shu Kawaguchi, Shigeru Mukai, Ken-Ichi Yoshikawa
  • Journal Title: American Journal of Mathematics Volume: 140 Issue: 6 Pages: 1471-1519
  • DOI: 10.1353/ajm.2018.0045
  • Peer Reviewed
• [Journal Article] Resultants and the Borcherds Φ-function (2018)
  • Author(s): Shu Kawaguchi, Shigeru Mukai, Ken-Ichi Yoshikawa
  • Journal Title: American Journal of Mathematics Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Analytic torsion for Borcea-Voisin threefolds (2017)
  • Author(s): K.-I. Yoshikawa
  • Journal Title: Progress in Mathematics Volume: 310 Pages: 279-361
  • DOI: 10.1007/978-3-319-49638-2_13
  • ISBN: 9783319496368, 9783319496382
  • Peer Reviewed
• [Journal Article] Analytic torsion for Borcea-Voisin threefolds (2017)
  • Author(s): Ken-Ichi Yoshikawa
  • Journal Title: Progress in Mathathematics Volume: 310 Pages: 279-361
  • Peer Reviewed
• [Journal Article] 対合付きK3曲面と解析的捩率 (2016)
  • Author(s): 吉川 謙一
  • Journal Title: 数学 Volume: 68 Pages: 225-245
  • NAID: 130007420317
  • Peer Reviewed
• [Presentation] Degeneration of Riemann surfaces and small eigenvalues of Laplacian (2021)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: Index Theory and Complex Geometry: Conference on Index Theory and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Quasi-pullback of certain Siegel modular forms and Borcherds products (2021)
  • Author(s): 吉川謙一
  • Organizer: 「保型形式，保型表現, ガロア表現とその周辺」RIMS共同研究（公開型）
  • Int'l Joint Research / Invited
• [Presentation] Degeneration of Riemann surfaces and small eigenvalues of Laplacian (2021)
  • Author(s): 吉川謙一
  • Organizer: Grauert theory and recent complex geometry (Grauert理論と最近の複素幾何)
  • Int'l Joint Research / Invited
• [Presentation] Degeneration of Riemann surfaces and small eigenvalues of Laplacian (2021)
  • Author(s): 吉川謙一
  • Organizer: India-Japan Web-Workshop on Vector Bundles and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Enriques 2n-folds and analytic torsion (2019)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: Discussion Meeting on Bundle 2019
  • Int'l Joint Research / Invited
• [Presentation] Enriques 2n-folds and analytic torsion (2018)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: Intercity Seminar in Arakelov Geometry 2018
  • Int'l Joint Research / Invited
• [Presentation] Enriques manifolds and analytic torsion (2018)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: 第13回代数・解析・幾何学セミナー
  • Int'l Joint Research / Invited
• [Presentation] Enriques manifolds and analytic torsion (2017)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: 第23回複素幾何シンポジウム
  • Int'l Joint Research / Invited
• [Presentation] Enriques manifolds and analytic torsion (2017)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: K3 Surfaces and Related Topics
  • Int'l Joint Research / Invited
• [Presentation] Holomorphic torsion invariants for K3 surfaces with involution and Borcherds products (2016)
  • Author(s): Ken-Ichi Yoshikawa
  • Organizer: Moduli spaces and modular forms
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach
  • Int'l Joint Research / Invited