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.

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    $\begingroup$ 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. $\endgroup$ – Remke Kloosterman Jun 17 '11 at 20:13
  • $\begingroup$ 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. $\endgroup$ – shenghao Jun 18 '11 at 14:34
  • $\begingroup$ 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. $\endgroup$ – shenghao Jun 19 '11 at 10:27

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 $H^{odd}(B) = 0$.

  • $\begingroup$ Does anyone know a source for this specific result (I don't see this specific result in GH or Grothendieck's paper on Chern classes)? $\endgroup$ – DZN Jun 18 '11 at 23:13
  • $\begingroup$ Any textbook. Griffiths-Harris, Fulton, etc. $\endgroup$ – Sasha Jun 19 '11 at 17:28

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