2023 Fiscal Year Final Research Report
Geometry of Veech surfaces
| Project/Area Number |
17K14184
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Shizuoka University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2024-03-31
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| Keywords | 平坦曲面 / Veech群 / 周期行列 |
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| Outline of Final Research Achievements |
This study aims to extend the properties known for Veech surfaces of genus 2 to general genera to clarify geometric properties of Veech surfaces.The first result is the clarification of the aspect of simple closed geodesics on hyperelliptic translation surfaces of general genus. This is an extention of the properties known in the case of genus 2.The second result is that we have indicated the periodic matrices of certain hyperelliptic translation surfaces. In particular, we gave an explicit indication in the case of genus 2.The third is the classification of hyperelliptic translation surfaces of genera 3 and 4 under certain conditions. In the case of genus 3, we found that there are two types of hyperelliptic translation surfaces depending on the conditions, and we extended this to the case of genus 4.
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| Free Research Field |
函数論
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| Academic Significance and Societal Importance of the Research Achievements |
1つ目の成果について，証明は種数2の場合とは異なる手法を用いた．種数0の場合の平坦曲面の新たな性質を解明しそれを利用しており，更なる応用が期待できる．2つ目の成果については，超楕円的平坦曲面を一般種数で更にパラメータ付きで扱っている.これまでの周期行列に関する先行研究は個別のリーマン面に対してのものであり，パラメータ付きで周期行列を扱ったことには意義がある．また種数2の場合には周期行列の各成分を多項式で与えていることも重要な点である．3つ目の成果についてはこれまで扱われてこなかった性質の研究である．種数3，4の場合を調べることで今後の更なる性質の解明の足掛かりとなることが期待できる．
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