Studies on (noncommutative) algebraic varieties via categorical points of view
| Project/Area Number |
16H05994
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| Research Category |
Grant-in-Aid for Young Scientists (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | Osaka University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥9,750,000 (Direct Cost: ¥7,500,000、Indirect Cost: ¥2,250,000)
Fiscal Year 2019: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2018: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2017: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000)
Fiscal Year 2016: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
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| Keywords | 導来圏 / 三角圏 / 極小モデル理論 / モジュライ / 非可換射影幾何学 / 非可換代数幾何学 / 半直交分解 / モジュライ空間 / 代数多様体 / 双有理幾何学 / 代数幾何学 / アーベル圏 / 代数多様体のGrothendieck環 |
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| Outline of Final Research Achievements |
I worked on several problems of noncommutative algebraic geometry (= a field where people study abelian or enhanced triangulated categories as geometric objects generalizing the category of coherent sheaves). We, Among others, I. proved that the moduli space classifying semiorthogonal decompositions of the derived category of coherent sheaves is an etale algebraic space; II. defined and confirmed the basics of the minimal model theory for b-boundary divisors; III. proved that the moduli stack of stable pointed curves are (almost) always rigid in the noncommutative sense; IV. gave general definition of noncommutative del Pezzo surfaces; V. proposed a few hypotheses concerning the relationship between derived equivalence of algebraic varieties and the additive invariants. We also found a few interesting examples.
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| Academic Significance and Societal Importance of the Research Achievements |
円や放物線のように「方程式=0」という形で表現される図形を代数多様体と呼び、数学内外の様々な分野と関係がある。代数多様体上には連接層というある種の線型な対象が(無数に)あるが、その全体の有り様(専門用語で圏)を研究することで代数多様体の本質に迫るのが非可換代数幾何学である。比較的若い分野であるためまだまだ基本的で重要なことがわかっていないが、例えば上記の成果Iはその一つを明らかにしたものである。また、Vは連接層の圏が代数多様体の本質をどこまで捉えているかという、非可換代数幾何学そのものの意義に関わる研究である。IVは非可換射影幾何学で研究されるべき対象を一気に増やしたという意義がある。
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Report
(5 results)
Research Products
(42 results)
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[Presentation] Noncommutative Hirzebruch surfaces2016
Author(s)
Shinnosuke Okawa
Organizer
Categorical and analytic invariants in Algebraic geometry 3
Place of Presentation
Higher School of Economics
Year and Date
2016-09-12
Related Report
Int'l Joint Research / Invited
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[Presentation] Noncommutative Hirzebruch surfaces2016
Author(s)
Shinnosuke Okawa
Organizer
School and Workshop on Homological Methods in Algebra and Geometry
Place of Presentation
African Institute for Mathematical Sciences Ghana
Year and Date
2016-08-01
Related Report
Int'l Joint Research / Invited
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[Presentation] On noncommutative Hirzebruch surfaces2016
Author(s)
Shinnosuke Okawa
Organizer
Non-commutative crepant resolutions, Ulrich Modules and generalizations of the Mckay correspondence
Place of Presentation
RIMS, Kyoto University
Year and Date
2016-06-13
Related Report
Int'l Joint Research / Invited
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