On symplectic automorphisms of a surface with genus two fibration and their action on $\mathrm{CH}_0$
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| Format: | Preprint |
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2025
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| _version_ | 1866915963352383488 |
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| author | Du, Jiabin Liu, Wenfei |
| author_facet | Du, Jiabin Liu, Wenfei |
| contents | Let $S$ be a complex smooth projective surface with a genus two fibration, and $\mathrm{Aut}_s(S)$ the group of symplectic automorphisms, fixing every holomorphic 2-forms (if any) on $S$. Based on the work of Jin-Xing Cai, we observe in this paper that, if $χ(\mathcal{O}_S)\geq 5$, then $|\mathrm{Aut}_s(S)|\leq 2$. Then we go on to verify, under some conditions, that $\mathrm{Aut}_s(S)$ acts trivially on the Albanese kernel $\mathrm{CH}_0(S)_{\mathrm{alb}}$ of the 0-th Chow group, which is predicted by a conjecture of Bloch and Beilinson. As a consequence, if an automorphism $σ\in \mathrm{Aut}(S)$ acts trivially on $H^{i,0}(S)$ for $0\leq i\leq 2$, then it also acts trivially on $\mathrm{CH}_0(S)_{\mathrm{alb}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On symplectic automorphisms of a surface with genus two fibration and their action on $\mathrm{CH}_0$ Du, Jiabin Liu, Wenfei Algebraic Geometry 14J50, 14J29, 14C15 Let $S$ be a complex smooth projective surface with a genus two fibration, and $\mathrm{Aut}_s(S)$ the group of symplectic automorphisms, fixing every holomorphic 2-forms (if any) on $S$. Based on the work of Jin-Xing Cai, we observe in this paper that, if $χ(\mathcal{O}_S)\geq 5$, then $|\mathrm{Aut}_s(S)|\leq 2$. Then we go on to verify, under some conditions, that $\mathrm{Aut}_s(S)$ acts trivially on the Albanese kernel $\mathrm{CH}_0(S)_{\mathrm{alb}}$ of the 0-th Chow group, which is predicted by a conjecture of Bloch and Beilinson. As a consequence, if an automorphism $σ\in \mathrm{Aut}(S)$ acts trivially on $H^{i,0}(S)$ for $0\leq i\leq 2$, then it also acts trivially on $\mathrm{CH}_0(S)_{\mathrm{alb}}$. |
| title | On symplectic automorphisms of a surface with genus two fibration and their action on $\mathrm{CH}_0$ |
| topic | Algebraic Geometry 14J50, 14J29, 14C15 |
| url | https://arxiv.org/abs/2501.10058 |