2023 Fiscal Year Final Research Report
Mathematics on Calabi-Yau manifolds and related topics
| Project/Area Number |
20K03530
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Multi-year Fund |
|---|
| Section | 一般 |
|---|
| Review Section |
Basic Section 11010:Algebra-related
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
|
|---|
| Project Period (FY) |
2020-04-01 – 2024-03-31
|
| Keywords | K3曲面 / エンリケス曲面 / コーブル曲面 / アーベル曲面 / ヤコビ多様体 / カラビ・ヤウ多様体 / リシュロー同種写像 / 正標数 |
|---|
| Outline of Final Research Achievements |
In the early 19th Century, Riemann introduced the notion of Riemann surface and around 1900 the Italian school developed the theory of classification of algebraic surfaces. In 1960's Kodaira established the rigorous theory of classification of algebraic surfaces over the complex number field. Then, Bombieri-Mumford constructed the theory of classification of algebraic surfaces over the algebraically closed field of positive characteristic. In our research, based on the theory of algebraic surface, we classified the Coble surfaces with finite automorphism group by using the configuration of nodal curves and determined the structure of finite automorphism groups, the number of moduli and the number of boundary components. We also investigated the structure of Richelot isogenies of Jacobian varieties of algebraic curves of genus 2 and 3.
|
| Free Research Field |
代数幾何学
|
| Academic Significance and Societal Importance of the Research Achievements |
代数幾何学の発展の流れに沿った研究であり、エンリケス曲面という代数曲面の分類理論上重要な位置を占める曲面の退化として現れるコーブル曲面に対して、標数2の代数的閉体上、自己同型群が有限の場合にはどのようなものが存在しうるかということに対する解答を与えるとともに、有限自己同型群の構造、各類のモジュライ数や境界の成分の数を決定した。また、種数2、3の代数曲線のヤコビ多様体のリシュロー同種写像の構造に関する結果を得たが、これは情報理論で現在活発に研究されている耐量子計算機暗号の理論と関係している。
|
