Global critical chart for local Calabi-Yau threefolds
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911983008219136 |
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| author | Kinjo, Tasuki Masuda, Naruki |
| author_facet | Kinjo, Tasuki Masuda, Naruki |
| contents | In this paper, we investigate Keller's deformed Calabi--Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $ω_Y$-torsor as a certain deformed $(n+1)$-Calabi--Yau completion of the derived category of $Y$.
As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, i.e., a Calabi--Yau threefold of the form $\mathrm{Tot}_C(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar--Vafa invariants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_10052 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Global critical chart for local Calabi-Yau threefolds Kinjo, Tasuki Masuda, Naruki Algebraic Geometry Category Theory Representation Theory In this paper, we investigate Keller's deformed Calabi--Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $ω_Y$-torsor as a certain deformed $(n+1)$-Calabi--Yau completion of the derived category of $Y$. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, i.e., a Calabi--Yau threefold of the form $\mathrm{Tot}_C(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar--Vafa invariants. |
| title | Global critical chart for local Calabi-Yau threefolds |
| topic | Algebraic Geometry Category Theory Representation Theory |
| url | https://arxiv.org/abs/2112.10052 |