2021 Fiscal Year Annual Research Report
Perverse sheaves and schobers
| Project/Area Number |
20H01794
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| Allocation Type | Single-year Grants |
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Bondal Alexey 東京大学, カブリ数物連携宇宙研究機構, 客員上級科学研究員 (00726408)
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| Co-Investigator(Kenkyū-buntansha) |
大川 新之介 大阪大学, 大学院理学研究科, 准教授 (60646909)
桑垣 樹 大阪大学, 大学院理学研究科, 助教 (60814621)
KAPRANOV MIKHAIL 東京大学, カブリ数物連携宇宙研究機構, 教授 (90746017)
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| Project Period (FY) |
2020-04-01 – 2025-03-31
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| Keywords | Derived categories / schobers / Floor theory / noncommutative / resolutions |
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| Outline of Annual Research Achievements |
The principal investigator A. Bondal developed the theory of noncommutative resolutions in the geometric and algebraic contexts. Algebraic resolutions were constructed via generalized noncommutative differential calculus for a collection of algebras and homomorphisms between them. Noncommutative resolutions for non-normal algebraic varieties were constructed in collaboration with co-Investigator S. Okawa by means of the universal fibered and cofibered squares.
Co-investigator M. Kapranov (in collaboration with V. Schechtman) explicitly described perverse sheaves on the Ran space of the complex line. The categorical interpretations of this construction was explored.
Co-investigator S. Okawa proved that the category of coherent right modules over a smooth noncommutative surface finite over its center is equivalent to a direct summand of the category of coherent sheaves of a smooth tame algebraic stack, which is canonically associated to it, thereby confirming that such nc surfaces are noncommutative geometric schemes in the sense of Orlov. The paper on this results is submitted to the electronic arxive.
As a byproduct of his research on sheaf-theoretic quantization co-investigator T.Kawasaki found a sheaf-theoretic version of the bounding cochain, which was known before in the context of Floer theory.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
COVID restrictions did not allow to implement the travel plans and to invite people collaborators from oversees.
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| Strategy for Future Research Activity |
We will develop the study of noncommutative resolutions via exact categories. We expect to obtain various schobers which govern the families of noncommutative resolutions of a category by means of varying exact structures on the category and considering the corresponding derived categories. We plan to apply this technique to constructing schobers of resolutions for finite dimensional algebras, as well as schobers of singularities of varieties.
We will work on the conjectural relationship between affine Weyl groups and polarizations of nc del Pezzo surfaces, and phantoms and quasi-phantom categories in positive characteristics.
Our new formalism of nonexact sheaf quantization at least enables us to formulate the sheaf theoretic side of the expected correspondence between sheaf quantization and Floer theory. We plan to explore the Floer side and the correspondence in the next year.
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