Timeline for The Gysin Sequence for an Associated Bundle over a Partial Flag Variety
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 9, 2013 at 11:21 | vote | accept | Peter Crooks | ||
| Sep 9, 2013 at 5:48 | answer | added | Geordie Williamson | timeline score: 3 | |
| Sep 8, 2013 at 14:54 | comment | added | Peter Crooks | Hi Allen, that is a good idea. I have added a question at the end. If Geordie posts his comment as an answer, I will definitely accept it. | |
| Sep 8, 2013 at 14:51 | history | edited | Peter Crooks | CC BY-SA 3.0 |
question added
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| Sep 8, 2013 at 6:39 | comment | added | Allen Knutson | It occurs to me that you don't have a question. Maybe you could ask one, such that Geordie's comment could become an answer. | |
| Sep 7, 2013 at 17:48 | comment | added | Geordie Williamson | Juteau studies this situation in detail (when P is the parabolic corresponding to the stabilizer of the highest root) in arxiv.org/abs/0704.3417. His main concert is to do it integrally, when the hard Lefschetz theorem only tells you part of the story. He finds a very satisfactory description of the middle cohomology in terms of the root system. | |
| Sep 7, 2013 at 16:01 | comment | added | Allen Knutson | If it comes from a regular dominant weight then it's a hyperplane class, and Hard Lefschetz will tell you it's injective up to middle dimension. The highest root usually isn't $P$-regular. Probably you'd want to think about fibering $G/P$ over $G/P'$ where the highest root is $P'$-regular, i.e. in the interior of the wall $(T^*_+)^{W_P}$ of the Weyl chamber. | |
| Sep 7, 2013 at 15:28 | comment | added | Peter Crooks | Thank you, that is very helpful! For my purposes, it suffices to know this up to sign. I'll now try to use the Gysin sequence to find $H^*$ of the circle bundle. It would be nice if taking the cup product with the first Chern class were injective (on sufficiently low-degree cohomology), but I think this will depend heavily on the weight. I'll need to think about this carefully. | |
| Sep 7, 2013 at 15:05 | comment | added | Allen Knutson | Definitely your Euler class definition (which I would rather call first Chern class, for no strongly arguable reason) gives a $W$-equivariant morphism from $(T^*)^{W_P}$ to $H^2(G/P)$, indeed to $H^2_G(G/P)$, so your guess must be right up to a multiple. For the inverse take your $G$-equivariant line bundle $L$ to $wt(L|_{P/P})$, so the multiple must be $\pm 1$, which is good enough I hope. | |
| Sep 7, 2013 at 12:29 | history | asked | Peter Crooks | CC BY-SA 3.0 |