Timeline for odd betti numbers of a projective bundle
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2011 at 16:00 | vote | accept | DZN | ||
| Jun 19, 2011 at 10:27 | comment | added | shenghao | In Grothendieck's Chern classes paper, "this specific" property you asked below is built into the axiom A1 (see p.5), and for singular cohom. he said this is well-known (see top of p.9). I don't know a precise reference, but I think it must be in some standard alg. top. book, maybe Bott-Tu? Anyway you may prove it using Leray to the map $f:P(E)\to B,$ which degenerates at $E_2$ by, for instance, Deligne's weight argument. Along the way you may need proper base change in topology. | |
| Jun 18, 2011 at 14:34 | comment | added | shenghao | In a reasonable cohomology theory where one can define Chern classes, one always has this relation between the cohom. of $P(E)$ and of $X$ (see e.g. Grothendieck's paper on Chern classes). For singular cohom. one can apply Kunneth formula. | |
| Jun 18, 2011 at 5:44 | vote | accept | DZN | ||
| Jun 18, 2011 at 6:10 | |||||
| Jun 18, 2011 at 4:22 | answer | added | Sasha | timeline score: 1 | |
| Jun 17, 2011 at 20:44 | history | edited | Yemon Choi |
removed superfluous tag
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| Jun 17, 2011 at 20:13 | comment | added | Remke Kloosterman | In Griffiths-Harris you find a description for the cohomology of a projective bundle. In particular, if h^i(B) is nonzero then h^i(P(E)) is also nonzero. | |
| Jun 17, 2011 at 20:03 | history | asked | DZN | CC BY-SA 3.0 |