Filtrations on the derived category of twisted K3 surfaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909652691714048 |
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| author | Chen, Zaiyuan Li, Zhiyuan Zhang, Ruxuan Zhang, Xun |
| author_facet | Chen, Zaiyuan Li, Zhiyuan Zhang, Ruxuan Zhang, Xun |
| contents | We introduce and study the Shen-Yin-Zhao filtration on derived categories of twisted K3 surfaces. A main contribution is the construction of a twisted Beauville-Voisin class $\mathfrak{o}_{\mathscr{X}} \in \operatorname{CH}_0(X)$ that extends fundamental results of O'Grady and Shen-Yin-Zhao \cite{OG13, SYZ20} to twisted settings. This class enables:
1. A derived equivalence-invariant filtration $\mathbf{S}_\bullet(\mathrm{D}^{(1)}(\mathscr{X}))$ preserved under Fourier-Mukai transforms,
2. A birational invariant filtration $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ on Bridgeland moduli spaces.
We prove $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ coincides with Voisin's filtration $\mathbf{S}^{\mathrm{BV}}_\bullet\operatorname{CH}_0$ (Theorem 1.4), providing a canonical candidate for the conjectural Beauville-Voisin filtration. Applications include Bloch's conjecture for (anti)-symplectic automorphisms and existence of algebraically coisotropic subvarieties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Filtrations on the derived category of twisted K3 surfaces Chen, Zaiyuan Li, Zhiyuan Zhang, Ruxuan Zhang, Xun Algebraic Geometry 14C25, 14F08, 14J28, 14J42 We introduce and study the Shen-Yin-Zhao filtration on derived categories of twisted K3 surfaces. A main contribution is the construction of a twisted Beauville-Voisin class $\mathfrak{o}_{\mathscr{X}} \in \operatorname{CH}_0(X)$ that extends fundamental results of O'Grady and Shen-Yin-Zhao \cite{OG13, SYZ20} to twisted settings. This class enables: 1. A derived equivalence-invariant filtration $\mathbf{S}_\bullet(\mathrm{D}^{(1)}(\mathscr{X}))$ preserved under Fourier-Mukai transforms, 2. A birational invariant filtration $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ on Bridgeland moduli spaces. We prove $\mathbf{S}^{\mathrm{SYZ}}_\bullet \operatorname{CH}_0$ coincides with Voisin's filtration $\mathbf{S}^{\mathrm{BV}}_\bullet\operatorname{CH}_0$ (Theorem 1.4), providing a canonical candidate for the conjectural Beauville-Voisin filtration. Applications include Bloch's conjecture for (anti)-symplectic automorphisms and existence of algebraically coisotropic subvarieties. |
| title | Filtrations on the derived category of twisted K3 surfaces |
| topic | Algebraic Geometry 14C25, 14F08, 14J28, 14J42 |
| url | https://arxiv.org/abs/2402.13793 |