Timeline for Borel–Weil–Bott for partial flag varieties
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 20, 2020 at 16:56 | vote | accept | Avi Steiner | ||
| Nov 20, 2020 at 14:13 | answer | added | jorge vargas | timeline score: 1 | |
| Nov 10, 2020 at 12:14 | comment | added | Will Sawin | @AviSteiner Because the fibers are connected (and have no coherent cohomology, if you're interested in the derived pushforward). This is a local question, so we can work locally on $G/P$, and in particular assume the line bundle is trivial - it's a question about $f_* \mathcal O_{G/B}$. We know this is trivial for a proper separable map with reduced connected fibers. | |
| Nov 10, 2020 at 3:31 | comment | added | Avi Steiner | @WillSawin Ok, I see why that answers my question. However, it’s not immediately clear to me why the natural map from a line bundle on $G/P$ to its pull-then-push is an isomorphism rather than just injective. (Though injectivity is enough to give me what I want) | |
| Nov 10, 2020 at 3:27 | comment | added | LSpice | @Jef's reference: Gruson, Sam, and Weyman - Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions. | |
| Nov 9, 2020 at 22:48 | comment | added | Jef | Section 2.3 of arxiv.org/abs/1203.2575 and references therein should be useful. | |
| Nov 9, 2020 at 22:10 | comment | added | Will Sawin | The cohomology of any equivariant line bundle on $G/P$ is equal to the cohomology of its pullback to $G/B$, which can be calculated using the usual Borel-Weil-Bott, by the Leray spectral sequence. | |
| Nov 9, 2020 at 22:01 | comment | added | Avi Steiner | @WillSawin How does that help? | |
| Nov 9, 2020 at 21:54 | comment | added | Will Sawin | Doesn't this follow from the usual Borel-Weil-Bott by noting that the (derived) pushforward from $G/B$ to $G/P$ of a line bundle pulled back from $G/P$ is again that line bundle? | |
| Nov 9, 2020 at 21:46 | history | asked | Avi Steiner | CC BY-SA 4.0 |