Timeline for Chern class of line bundle inducing a principal polarisation
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Jul 10, 2012 at 13:20 | comment | added | user19475 | "It doesn't depend on $f$" seems to be wrong: Take $f = \mathrm{id}$ and $f = 0$. | |
| Jun 29, 2012 at 21:07 | comment | added | Damian Rössler | @Timo Keller: see for instance Faltings-Chai, "Degeneration of abelian varieties", chap. I, p. 3-4 (esp. middle of p. 4). It doesn't depend on $f$ because, up to $2$-torsion line bundles, two symmetric line bundles, which are algebraically equivalent, must be isomorphic (because the only symmetric line bundle, which is algebraically equivalent to $0$ is a $2$-torsion line bundle). | |
| Jun 28, 2012 at 19:39 | comment | added | user19475 | Thank you, Damian. How does one prove your first equation? Why doesn't it depend on $f$? | |
| Jun 28, 2012 at 10:06 | comment | added | Francesco Polizzi | Well, up to a translation (which does not change the numerical class) one can assume that $\mathcal{L}$ is symmetric | |
| Jun 28, 2012 at 6:11 | comment | added | Damian Rössler | If $L$ is symmetric and ample then $(f,c_{\cal L})^*{\cal P}_A\simeq L^{\otimes 2}$ so ${\rm deg}((f,c_{\cal L})^*{\cal P}_A\otimes L^{\otimes -2})$ is independent of the choice of $\cal L$, provided $\cal L$ is symmetric. | |
| Jun 27, 2012 at 12:49 | history | edited | user19475 | CC BY-SA 3.0 |
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| Jun 26, 2012 at 18:19 | comment | added | Francesco Polizzi | @Will: you are right, it is not completely obvious | |
| Jun 26, 2012 at 17:48 | comment | added | Will Sawin | @Francesco: What if there is more than one principal polarization? For instance, a product of two isomorphic elliptic curves will always have many principal polarizations. | |
| Jun 26, 2012 at 14:03 | comment | added | Francesco Polizzi | Is it not obvious? It seems to me that the quantity $\mathrm{deg}(((f,c_\mathcal{L})^*\mathcal{P}_A \cup c_1(\mathcal{L})) \in \mathbf{Z}$ only depends on the type of the polarization, i.e. on the numerical equivalence class. So it is independent on the actual choice of the line bundle in its numerical class. Am I missing something? | |
| Jun 26, 2012 at 13:12 | history | edited | user19475 | CC BY-SA 3.0 |
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| Jun 26, 2012 at 13:07 | comment | added | user19475 | I will extend my question in a minute. | |
| Jun 26, 2012 at 12:47 | comment | added | Jason Starr | Dear Timo -- Could you say more precisely what you are looking for? | |
| Jun 26, 2012 at 12:25 | history | asked | user19475 | CC BY-SA 3.0 |