Timeline for Extensions for a short exact sequence on Grassmannians
Current License: CC BY-SA 4.0
9 events
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| Dec 9, 2021 at 17:13 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`
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| Dec 9, 2021 at 15:19 | answer | added | Jake Levinson | timeline score: 2 | |
| Dec 9, 2021 at 13:06 | answer | added | Vít Tuček | timeline score: 0 | |
| Dec 9, 2021 at 13:06 | history | edited | Bobech | CC BY-SA 4.0 |
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| Dec 9, 2021 at 12:53 | comment | added | Frank | You can do this using the Borel-Weil-Bott theorem and a bit of representation theory, as abx suggests. The bundle in question should be decomposable, and you can obtain such a decomposition using the Schurrings package of Macaulay2. After getting a decomposition, you can use BWB to compute the cohomology groups of each factor and take the sum. Have a look at proposition 3.1 here for some similar computations arxiv.org/pdf/2008.05162.pdf . After doing a few cases for small k,n, you can hopefully try to get a general picture for the decomposition of your bundle. | |
| Dec 9, 2021 at 12:00 | history | edited | Bobech | CC BY-SA 4.0 |
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| Dec 9, 2021 at 11:38 | comment | added | Bobech | I have computed the weights in general (it is a mess) and in the particular case of $G(2,4)$. There is at least one non dominant weight, in particular the $H^0$ will be zero, but it does not give me information about $H^1$ | |
| Dec 9, 2021 at 11:18 | comment | added | abx | These are homogeneous vector bundles, so in principle their cohomology is computable — I don't know if this is doable in practice. The standard reference is Cohomology of vector bundles and syzygies by Jerzy Weyman (CUP). | |
| Dec 9, 2021 at 10:19 | history | asked | Bobech | CC BY-SA 4.0 |