odd betti numbers of a projective bundle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T16:46:50Z http://mathoverflow.net/feeds/question/68086 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68086/odd-betti-numbers-of-a-projective-bundle odd betti numbers of a projective bundle DZN 2011-06-17T20:03:08Z 2011-06-18T04:22:10Z <p>I would like to know if the odd Betti numbers of a projective bundle P(E) for some vector bundle E over say a compact complex smooth algebraic variety B are zero just as in the case for ordinary projective spaces over Spec(k), or more generally how to generalize standard calculations of the cohomology of projective space to projective bundles.</p> http://mathoverflow.net/questions/68086/odd-betti-numbers-of-a-projective-bundle/68118#68118 Answer by Sasha for odd betti numbers of a projective bundle Sasha 2011-06-18T04:22:10Z 2011-06-18T04:22:10Z <p>If $E$ is of rank $r$ then $H^i(P_B(E)) = \sum_{t = 0}^{r-1} H^{i-2t}(B)$ (where the summands with negative $i - 2t$ are omitted). So $H^{odd}(P_B(E)) = 0$ if and only if $H^{odd}(B) = 0$.</p>