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Let $\mathcal{E}$ be a projective bundle of rank $r$ over a smooth complex quasi projective variety $B$, and form its associated projective bundle $\chi :=\mathbb{P}(\mathcal{E})$. Let $\pi : \chi \rightarrow B$ denote the projection map. It is known that the Chow ring and the cohomology ring of $\chi$ are given by

$CH^*(\chi)=\dfrac{CH^*(B)[h]}{f}$,

$H^*(\chi, \mathbb{Q})=\dfrac{H^*(B, \mathbb{Q})[h]}{f}$,

where $f= h^r+\pi^* c_1(\mathcal{E})h^{r-1}+ ...+ \pi^*c_r(\mathcal{E})$ with $c_i \in CH^i(B)$ the $i$-th Chern class of the vector bundle $\mathcal{E}$.

Now, the relative Kunneth formula also give us

$H^*(\chi \times_B \chi ,\mathbb{Q})=\dfrac{H^*(B,\mathbb{Q})[h_1, h_2]}{f_1,f_2}$, with $f_i:=f(h_i)$.

(Does such a formula also hold for the Chow ring?)

What I am looking for is a decomposition for the cohomology class of the relative diagonal $\Delta_{\chi}\in H^{2r-2}(\chi\times_B\chi,\mathbb{Q})$ accordingly to the decomposition above. I guess it is possible to describe it in terms of the Chern classes or some other cohomology classes attached to $\mathcal{E}$, am I wrong?

Thanks.

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  • $\begingroup$ Did you try writing this down for yourself? For instance, did you consider the top Chern class of the tensor product of the pullback by the first projection of the dual of the tautological subbundle of $\pi^*\mathcal{E}$ and the pullback by the second projection of the tautological quotient bundle of $\pi^*\mathcal{E}$? $\endgroup$
    – Jason Starr
    Commented May 28, 2016 at 16:19
  • $\begingroup$ As I understand you mean that the diagonal coincide with that top Chern class. Is this an obvious fact ? $\endgroup$
    – S. S.
    Commented May 28, 2016 at 18:02
  • $\begingroup$ I do not believe this is an "obvious fact". However, it does follow in a relatively straightforward way from the fundamental facts about Chern classes such as the Whitney sum formula, compatibility with pullbacks, and representability of Chern classes by those degeneracy loci that are "sufficiently transverse". In that sense, I think this is a good question to bear in mind and to work on as one learns the basic theory. $\endgroup$
    – Jason Starr
    Commented May 30, 2016 at 19:30
  • $\begingroup$ Thank you very much Jason, I have been able to understand why this is true. Does this characterization of the relative diagonal as degeneracy locus of a vector bundle admit generalizations to other families (e.g. smooth projective families of curves) ? Of course we will not have these tautological bundles, but I wonder whether it could be possible to construct some other vector bundle which plays a similar role in the description of the relative diagonal. $\endgroup$
    – S. S.
    Commented Jun 1, 2016 at 19:09
  • $\begingroup$ I am not aware of such a description of the relative diagonal for families of curves. In particular, inside the moduli space $\overline{\mathcal{M}}_{g,2}$ of $2$-pointed, stable curves, I do not know any general divisor class relation relating the "diagonal class" $\Delta_{(0,\{1,2\}),(g,\emptyset)}$ to some other linear combination of "tautological divisor classes". $\endgroup$
    – Jason Starr
    Commented Jun 1, 2016 at 19:22

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