Timeline for cohomology of flag variety
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10 events
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| Oct 22, 2018 at 8:55 | comment | added | wky | Here is a more refined question. Given that we know there is no higher cohomology in the above example, then there must be another weight in $S^2(\mathfrak{n}^*)$ that 'cancels' the $H^1$ given by the weight $(-2,2,0)$. And $(-2,1,1)$ seems to play the role that its $H^0$ will cancel out the $H^1$ of $(-2,2,0)$. How can we see this cancellation on the level $B$-modules? Or more generally, given a finite-dimensional $B$ module, can we find its cohomology explicitly? | |
| S Jul 30, 2018 at 13:21 | history | suggested | CommunityBot | CC BY-SA 4.0 |
Anderson --> Andersen
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| Jul 30, 2018 at 12:35 | review | Suggested edits | |||
| S Jul 30, 2018 at 13:21 | |||||
| Jul 30, 2018 at 12:34 | comment | added | dhy | Yep - that decomposition is fine (and works on the level of vector bundles). | |
| Jul 30, 2018 at 12:27 | comment | added | wky | Thanks a lot! I suspected the action on $N$ plays a role, and it indeed does. Just another question: I suppose the $B$-action on $S^{\bullet}(\mathfrak{n}^*)$ preserves the grading, so that one has $$H^i(G/B, S^{\bullet}(\mathfrak{n}^*)) = \bigoplus_k H^i(G/B, S^k(\mathfrak{n}^*))?$$ | |
| Jul 30, 2018 at 11:55 | comment | added | dhy | Incidentally, Anderson-Jantzen is mainly about the positive characteristic case. If you are interested only in the characteristic zero case, I think this theorem goes back to Kostant. In fact it also follows from the Grauert-Riemenschneider theorem applied to the Springer resolution (I am not sure who first observed this.) | |
| Jul 30, 2018 at 11:53 | comment | added | dhy | The answer to your question is no, as the vector bundle corresponding to $S^2(\mathfrak{n}^*)$ does not split into line bundles. This is because $S^2(\mathfrak{n}^*)$ does not split into a sum of one-dimensional representations as a $B$-representation ($B$ is not semisimple.) Instead, what you get is a filtration of your vector bundle by line bundles. | |
| Jul 30, 2018 at 11:23 | history | edited | j.c. | CC BY-SA 4.0 |
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| Jul 30, 2018 at 11:02 | review | First posts | |||
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| Jul 30, 2018 at 10:56 | history | asked | user127163 | CC BY-SA 4.0 |