2022 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 京都大学, 理学研究科, 准教授 (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 | resolution / coherent sheaf / perverse sheaf / Fukaya category |
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| Outline of Annual Research Achievements |
A criterion for a finite dimensional algebra to be quasi-hereditary is given in terms of a pair of exceptional collections of modules over the algebra.
Noncommutative resolutions of reduced curves were studied via fibered-cofibered squares of curves. Nocommutative resolutions of some finite length schemes were constructed via null categories of birational morphisms of smooth surfaces. It was investigated how to reconstruct a normal surface from the category of reflexive sheaves on it.
A version of the Riemann-Hilbert correspondence in the presence of the Planck parameter is proven.
A twist-closed dg-enhancement for the category of restricted objects in the derived category of coherent sheaves on noncompact complex-analytic manifolds was constructed via dbar-superconnections.
For noncommutative surfaces which are finite over their centers Artin stacks were constructed which are Morita equivalent to the noncommutative surfaces up to taking direct summands.
A generalization of the concept of spherical functors, which was named N-spherical functor and which describes N-periodic semi-orthogonal decomposition was developed. This allowed us to categorify Euler's continuants in the theory of continued fractions.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
The progress of the work on the research project is good. Noncommutative resolutions for curves were investigated from the point of view of fibered-cofibered squares of curves. A new interesting class of resolutions for 0-dimensional schemes was constructed via the null-categories of birational morphisms of smooth surfaces. It is proven that every restricted object of the directed category of coherent sheaf on noncompact manifold allows a presentation via a dbar-superconnection, thus giving a way to construct moduli spaces of this kind of objects. It was shown that the compactified moduli space of weighted projective lines is endowed with the sheaf of abelian categories of finite global dimension.
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| Strategy for Future Research Activity |
We plan to arrange a workshop in June 2023, where we invite the members of the team, our collaborators and leading experts working on the subject of the research project to give talks on their research and to exchange knowledge within our group and with the experts. We plan to develop the study of noncommutative resolutions and relevant schobers for 0-dimensional schemes, curves and surfaces. We wilI construct microlocal categories over Novikov rings, which should be the sheaf-theoretic counterpart of Fukaya categories over Novikov rings. We will study how to compute Efimov's categorical punctured neighborhood. We will describe how to reconstruct the normal surface from the category of reflexive sheaves on it.
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