Timeline for Fixed-point free holomorphic involutions
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 13, 2022 at 11:24 | comment | added | Jason Starr | "Is there a way to tell which hypersurfaces in $\mathbb{P}^n$ have such an involution?" For all $n\geq 4$, there is no such hypersurface. For $n\geq 4$, the restriction map on Picard groups from $\mathbb{P}^n$ to the hypersurface is an isomorphism. Thus, every automorphism extends to $\mathbb{P}^n$. The fixed locus of the involution of $\mathbb{P}^n$ equals two linear subspaces, at least one of which has dimension $>1$. The intersection of this linear subspace with the hypersurface is nonempty. | |
| Jul 12, 2022 at 19:56 | answer | added | abx | timeline score: 7 | |
| Jul 12, 2022 at 19:26 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |
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| Jul 12, 2022 at 14:16 | comment | added | Mohammad Farajzadeh-Tehrani | Thanks, Jason. That's helpful. Is there a way to tell which hypersurfaces in $\mathbb{P}^n$ have such an involution? (or better said, for what $d\geq 0$, there is a degree $d$ hypersurface in $\mathbb{P}^n$ admitting such an involution. | |
| Jul 12, 2022 at 2:16 | history | became hot network question | |||
| Jul 11, 2022 at 20:45 | answer | added | Francesco Polizzi | timeline score: 4 | |
| Jul 11, 2022 at 20:44 | comment | added | Sasha | In the opposite direction --- if $Y$ is a variety such that $\pi_1(Y)$ has a surjection onto $\mathbb{Z}/2$, the corresponding etale double covering $X$ of $Y$ has such an involution. | |
| Jul 11, 2022 at 18:59 | comment | added | Moishe Kohan | Many compact ball quotients admit fixed-point free holomorphic involutions. | |
| Jul 11, 2022 at 18:40 | comment | added | Jason Starr | I looked up that comment. There is no fixed-point-free involution acting on a hyper-Kaehler manifold whose (complex) dimension is divisible by $4$, e.g., there are such actions on K3 surfaces (with quotient an Enriques surface), but not on $4$-dimensional hyper-Kaehler manifolds. | |
| Jul 11, 2022 at 18:33 | comment | added | Jason Starr | Given one such variety $X$, for every variety $Y$, the product $X\times Y$ is another such variety. Starting from $X$ a curve of genus $g>0$, you can produce examples in every dimension $n$, e.g., $X\times \mathbb{P}^{n-1}$. I wrote a bit more about this once in a comment to an MO post by user abx . . . | |
| Jul 11, 2022 at 18:13 | history | asked | Mohammad Farajzadeh-Tehrani | CC BY-SA 4.0 |