Timeline for Purity of vanishing cycle for proper scheme over DVR with smooth generic fiber
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Aug 27, 2019 at 20:21 | history | bounty ended | Zhiyu | ||
| Aug 23, 2019 at 21:44 | comment | added | Will Sawin | @user45878 Possibly one can deduce it from Poincare duality (reducing it to giving a lower bound on the weights) and the fact that the Frobenius eigenvalues on the compactly supported cohomology of a variety must be integral (hence have weight at least zero). In the geometric setting, you can spread out to a punctured curve and compute the compactly supported cohomology of the total space, and you'll see the local monodromy invariant parts at the puncture show up. | |
| Aug 23, 2019 at 21:21 | comment | added | Pol van Hoften | Is there an easy way to see that the weights must be bounded by 2n (say without the weight spectral sequence)? | |
| Aug 23, 2019 at 20:58 | comment | added | Will Sawin | @sawdada Another method is to use purity considerations - we can take $E$ to be regular, meaning the pushforward is a pure complex, which forces skyscraper components to have the expected weight. But if you like the Tate curve, one could write down a $\mathbb Z/\ell$-torsor over the special fiber, extend it to the generic point, and note that it remains a nontrivial torsor. In fact you should obtain the covering of $\mathbb G_m/q$ by $\mathbb G_m / q^\ell$. | |
| Aug 23, 2019 at 20:43 | comment | added | Zhiyu | Thanks for a good example by taking product, is there an easy way to see $H^1(E_s) \rightarrow H^1(E_{\eta})$ is injective (without applying Picard-Lefschetz formula)? The Galois action on $H^1(E_s)$ and $H^1(E_{\eta})$ is easy to compute using Tate curve and restriction exact sequence. | |
| Aug 23, 2019 at 19:42 | history | answered | Will Sawin | CC BY-SA 4.0 |