Moduli of sheaves on fourfolds as derived Lagrangian intersections
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arXiv
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| Format: | Preprint |
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2024
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| author | Adhikari, Nachiketa Shi, Yun |
| author_facet | Adhikari, Nachiketa Shi, Yun |
| contents | We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any $n$-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme.
We also give a separate proof of our main result in the case where the local Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree $(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00727 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moduli of sheaves on fourfolds as derived Lagrangian intersections Adhikari, Nachiketa Shi, Yun Algebraic Geometry We show that any $(-2)$-shifted symplectic derived scheme $\textbf{X}$ (of finite type over an algebraically closed field of characteristic zero) is locally equivalent to the derived intersection of two Lagrangian morphisms to a $(-1)$-shifted symplectic derived scheme which is the $(-1)$-shifted cotangent stack of a smooth classical scheme. This leads to the possibility of the following viewpoint that is, at least to us, new: any $n$-shifted symplectic derived scheme can be obtained, locally, by repeated derived Lagrangian intersections in a smooth classical scheme. We also give a separate proof of our main result in the case where the local Darboux atlas cdga for $\textbf{X}$ has an even number of generators in degree $(-1)$; in this case we strengthen the result by showing that $\textbf{X}$ is in fact locally equivalent to the derived critical locus of a shifted function, which we've been told is a folklore result in the field. We indicate the implications of this for derived moduli stacks of sheaves on Calabi-Yau fourfolds by spelling out the case when the fourfold is $\mathbb{C}^4$. |
| title | Moduli of sheaves on fourfolds as derived Lagrangian intersections |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2403.00727 |