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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2311.16287 |
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| _version_ | 1866916080214081536 |
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| author | Baird, Thomas John |
| author_facet | Baird, Thomas John |
| contents | Let $S$ be a smooth, quasi-projective complex surface with complex symplectic form $ω\in H^0(S, K_S)$. This determines a symplectic form $ω_n$ on the Hilbert scheme of points $S^{[n]}$ for $n \geq 1$. Let $τ$ be an anti-symplectic involution of $(S,ω)$: an order two automorphism of $S$ such that $ τ^*ω=-ω$. Then $τ$ induces an anti-symplectic involution on $(S^{[n]},ω_n)$ and the fixed point set $(S^{[n]})^τ$ is a smooth Lagrangian subvariety of $S^{[n]}$. In this paper, we calculate the mixed Hodge structure of $H^*( (S^{[n]})^τ; \mathbb{Q})$ in terms of the mixed Hodge structures of $H^*( S^τ;\mathbb{Q})$ and of $H^*( S / τ; \mathbb{Q})$. We also classify the connected components of $(S^{[n]})^τ$ and determine their mixed Hodge structures. Our results apply more generally whenever $S$ is a smooth quasi-projective surface, and $τ$ is an involution of $S$ for which $S^τ$ is a curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_16287 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cohomology of fixed point sets of anti-symplectic involutions in the Hilbert scheme of points on a surface Baird, Thomas John Algebraic Geometry 14C30 Let $S$ be a smooth, quasi-projective complex surface with complex symplectic form $ω\in H^0(S, K_S)$. This determines a symplectic form $ω_n$ on the Hilbert scheme of points $S^{[n]}$ for $n \geq 1$. Let $τ$ be an anti-symplectic involution of $(S,ω)$: an order two automorphism of $S$ such that $ τ^*ω=-ω$. Then $τ$ induces an anti-symplectic involution on $(S^{[n]},ω_n)$ and the fixed point set $(S^{[n]})^τ$ is a smooth Lagrangian subvariety of $S^{[n]}$. In this paper, we calculate the mixed Hodge structure of $H^*( (S^{[n]})^τ; \mathbb{Q})$ in terms of the mixed Hodge structures of $H^*( S^τ;\mathbb{Q})$ and of $H^*( S / τ; \mathbb{Q})$. We also classify the connected components of $(S^{[n]})^τ$ and determine their mixed Hodge structures. Our results apply more generally whenever $S$ is a smooth quasi-projective surface, and $τ$ is an involution of $S$ for which $S^τ$ is a curve. |
| title | Cohomology of fixed point sets of anti-symplectic involutions in the Hilbert scheme of points on a surface |
| topic | Algebraic Geometry 14C30 |
| url | https://arxiv.org/abs/2311.16287 |