Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913555573374976 |
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| author | Casalaina-Martin, Sebastian Hu, Xianyu Lin, Xun Zhang, Shizhuo Zhang, Zheng |
| author_facet | Casalaina-Martin, Sebastian Hu, Xianyu Lin, Xun Zhang, Shizhuo Zhang, Zheng |
| contents | Let $Y$ be a cubic threefold with a non-Eckardt type involution $τ$. Our first main result is that the $τ$-equivariant category of the Kuznetsov component $\mathcal{K}u_{\mathbb{Z}_2}(Y)$ determines the isomorphism class of $Y$ for general $(Y,τ)$. We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction due to Alexander Perry), a Bridgeland moduli theoretical one (using equivariant stability conditions), and a Chow theoretical one (using some techniques in [kuznetsovnonclodedfield2021]).The remaining part of the paper is devoted to proving an equivariant infinitesimal categorical Torelli for non-Eckardt cubic threefolds $(Y,τ)$. To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category $D^b(X)$ of a smooth projective variety $X$ and on the Hochschild cohomology of a semi-orthogonal component of $D^b(X)$. Another key ingredient is a generalization of a result in [macri2009infinitesima] which shows that the twisted Hochschild-Kostant-Rosenberg isomorphism is compatible with the actions on the Hochschild cohomology and on the singular cohomology induced by an automorphism of $X$. In appendix, we prove an equivariant categorical Torelli theorem for arbitrary cubic threefold with a geometric involution under a natural assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_20554 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components Casalaina-Martin, Sebastian Hu, Xianyu Lin, Xun Zhang, Shizhuo Zhang, Zheng Algebraic Geometry 14F05, 14J45, 14D20, 14D23 Let $Y$ be a cubic threefold with a non-Eckardt type involution $τ$. Our first main result is that the $τ$-equivariant category of the Kuznetsov component $\mathcal{K}u_{\mathbb{Z}_2}(Y)$ determines the isomorphism class of $Y$ for general $(Y,τ)$. We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction due to Alexander Perry), a Bridgeland moduli theoretical one (using equivariant stability conditions), and a Chow theoretical one (using some techniques in [kuznetsovnonclodedfield2021]).The remaining part of the paper is devoted to proving an equivariant infinitesimal categorical Torelli for non-Eckardt cubic threefolds $(Y,τ)$. To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category $D^b(X)$ of a smooth projective variety $X$ and on the Hochschild cohomology of a semi-orthogonal component of $D^b(X)$. Another key ingredient is a generalization of a result in [macri2009infinitesima] which shows that the twisted Hochschild-Kostant-Rosenberg isomorphism is compatible with the actions on the Hochschild cohomology and on the singular cohomology induced by an automorphism of $X$. In appendix, we prove an equivariant categorical Torelli theorem for arbitrary cubic threefold with a geometric involution under a natural assumption. |
| title | Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components |
| topic | Algebraic Geometry 14F05, 14J45, 14D20, 14D23 |
| url | https://arxiv.org/abs/2405.20554 |