Serre-invariant stability conditions and Ulrich bundles on cubic threefolds
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866918367626002432 |
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| author | Feyzbakhsh, Soheyla Pertusi, Laura |
| author_facet | Feyzbakhsh, Soheyla Pertusi, Laura |
| contents | We prove a general criterion which ensures that a fractional Calabi--Yau category of dimension $\leq 2$ admits a unique Serre-invariant stability condition, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component $\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the known stability conditions on $\text{Ku}(X)$ are invariant with respect to the action of the Serre functor and thus lie in the same orbit with respect to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As an application, we show that the moduli space of Ulrich bundles of rank $\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macrì and Stellari. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_13549 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Serre-invariant stability conditions and Ulrich bundles on cubic threefolds Feyzbakhsh, Soheyla Pertusi, Laura Algebraic Geometry We prove a general criterion which ensures that a fractional Calabi--Yau category of dimension $\leq 2$ admits a unique Serre-invariant stability condition, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component $\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the known stability conditions on $\text{Ku}(X)$ are invariant with respect to the action of the Serre functor and thus lie in the same orbit with respect to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As an application, we show that the moduli space of Ulrich bundles of rank $\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macrì and Stellari. |
| title | Serre-invariant stability conditions and Ulrich bundles on cubic threefolds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2109.13549 |