| Project/Area Number |
25287007
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Partial Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Hiroshima University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
山田 裕史 熊本大学, 大学院先端科学研究部(理), 教授 (40192794)
與倉 昭治 鹿児島大学, 理工学域理学系, 教授 (60182680)
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| Project Period (FY) |
2013-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥8,840,000 (Direct Cost: ¥6,800,000、Indirect Cost: ¥2,040,000)
Fiscal Year 2017: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2016: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2015: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2013: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
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| Keywords | モチーフ / K環 / 有限次元性 / 代数幾何 / 代数学 / トーリック多様体 / Chow級数 / 有理性、有限性 |
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| Outline of Final Research Achievements |
A circle can be described by the equation "x squared plus y squared equals 1”. When we use only two numbers, namely 0=”even numbers” and 1=”odd numbers”, then the set of two points {(even, odd), (odd, even)} is a “shape” described by the same equation, and behaves “like” a circle. Motif theory explains this phenomenon as follows: There is the Ideal Circle (motif of a circle) and the real circle, and the two points as above, are the shadows of the same ideal circle, hence behave similarly. In this research project, we try to prove the conjecture that the Ideal Algebraic Varieties (Motives of algebraic varieties) are finite dimensional (Not too large, and hence computable). As a result Joseph Ayoub proved the finite dimensionality, solving many big problems of motif theory.
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| Academic Significance and Societal Importance of the Research Achievements |
図形をあらわす方程式を別の「数」で考えて図形の情報を引き出す、というアイデアは１９５０年台のWeil によるもので、そのアイデアに触発されて２０世紀後半にエタールコホモロジーを含む代数幾何及び数論幾何の大きな進展があった。しかしその証明は、方程式（＝モチーフ、図形の神様）の本体ではなく別の影（別の数での図形情報）を詳しくみるもので、モチーフ本体はもっと複雑かもしれなかった。今回モチーフの有限次元性が証明されたことで、モチーフ本体の複雑さ（次元）はコホモロジーの複雑さ（次元）と同じであることがわかり、今後様々な問題がコホモロジーの計算に帰着されることになるはずである。
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