Studies on noncommutative algebraic geometry and generalized complex geometry
| Project/Area Number |
16K13746
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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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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| Co-Investigator(Kenkyū-buntansha) |
後藤 竜司 大阪大学, 理学研究科, 教授 (30252571)
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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 |
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 非可換代数幾何学 / AS正則代数 / 一般化された複素構造 / 導来圏 / 非可換射影空間 / 正則Poisson構造 / Hirzebruch曲面 / 非可換del Pezzo曲面 / ポアソン構造 |
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| Outline of Final Research Achievements |
We generalized the classes of noncommutative algebras corresponding to noncommutative projective planes and noncommutative quadrics so that any noncommutative del Pezzo surface correspond to one of these classes of algebras. Also we partially generalized the correspondence between those algebras and commutative algebro-geometric data.
We gave a conjectural HKR isomorphism for Deligne-Mumford stacks, and confirmed it in several interesting examples. Inspired by this, we defined the notion of generalized complex orbifold.
We found a class of noncommutative graded algebras which yield noncommutative weighted projective 3-spaces P(1,1,1,2), and found the corresponding Poisson geometry on the commutative P(1,1,1,2). These are related to the geometry of noncommutative/Poisson del Pezzo surfaces of degree 2.
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| Academic Significance and Societal Importance of the Research Achievements |
非可換射影幾何学、一般化された複素幾何のどちらにとっても、非可換/一般化されたdel Pezzo曲面は最も重要で基本的な例である。非可換射影幾何学においては射影平面と2次曲面の場合を除いて基礎づけに不満があったが、それを補う形で一般の非可換del Pezzo曲面を定義するようなAS正則代数のクラスを定義することができた。また、orbifoldの非可換変形に関する理解が進み、思った以上に豊かな現象が起きていることがわかった。さらに、Poisson幾何との対応を参照することで、高次元の新たなAS正則代数のクラスが発見できた。これらは今後研究対象となるべきものである。
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Report
(5 results)
Research Products
(39 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, Russia
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
京都大学数理解析研究所
Year and Date
2016-06-13
Related Report
Int'l Joint Research / Invited
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