Timeline for What is the relation between vector bundles on a manifold and grassmanians?
Current License: CC BY-SA 2.5
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| Jul 21, 2010 at 0:21 | history | edited | Anweshi | CC BY-SA 2.5 |
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| Jul 21, 2010 at 0:03 | comment | added | BCnrd | Anweshi, it's fine; I just thought it might freak out the OP to be told that one should learn about Quot schemes etc. to understand Grassmannians (which of course isn't really necessary). | |
| Jul 20, 2010 at 21:22 | comment | added | Anweshi | @BCnrd: When I wrote "component of the Hilbert scheme", what I had in mind was the following. If $E$ is a locally free sheaf of rank $n$ and if the polynomial $P = n -m$, then $Quot_{E, P}$ is the same as $Gr(n,m)$. About your other objections: I just wanted to enlighten the OP about the Hilbert scheme, leading him on to moduli problems. If you(or others) feel that this answer is too uninformative to be kept, please leave one more comment and I would be happy to delete it. | |
| Jul 20, 2010 at 20:29 | comment | added | BCnrd | Anweshi, why do you say that a Grassmannian is a component of a Hilbert scheme, when to the contrary (as you note later) Hilbert schemes with a fixed Hilbert polynomial are naturally found inside of Grassmannians? Anyway, I disagree with the advice to leave the manifold setting, since the basics of Grassmannians work equally well in all geometric settings (smooth, real-analytic, complex-analytic, rigid-analytic, algebraic, etc.), so one can study the question in whatever geometric setting is most familiar. Bringing in Hilbert schemes here seems like a long detour for no purpose. | |
| Jul 20, 2010 at 19:40 | history | answered | Anweshi | CC BY-SA 2.5 |