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Let $X$ be a smooth projective rational variety over $\mathbb{C}$, and let $\pi:Y\rightarrow X$ be a principal projective bundle with fibers isomorphic to $SL(n,\mathbb{C})/P$, where $P$ is a parabolic subgroup. Consider the sheaf $\mathbb{G}_m$ on $Y$ in e'tale topology.

I'm mainly interested in the cohomologies of the higher direct images. Can we describe the higher direct images $R^i\pi_*(\mathbb{G}_m)$ on $X$? For example, is $\pi_*\mathbb{G}_m$ a locally constant sheaf on $X$?

Any help would be appreciated.

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  • $\begingroup$ Could you reproduce or give a reference for the definition of principal projective bundles? $\endgroup$
    – მამუკა ჯიბლაძე
    Sep 7, 2021 at 19:38
  • $\begingroup$ A principal bundle is étale-locally trivial, so for most properties of $R^i \pi_* (\mathbb G_m)$ you may as well assume $Y$ is a trivial bundle, i.e. $\pi \colon SL_n/P \times X \to X$ is the projection. Certainly in this case, and in general, $\pi_* \mathbb G_m \cong \mathbb G_m$ is not a locally constant sheaf. $\endgroup$
    – Will Sawin
    Sep 7, 2021 at 22:03
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    $\begingroup$ By a principal projective bundle, I mean a principal $PGL_(r+1)$-bundle for some r, with fibers $\mathbb{P}^r$. $\endgroup$
    – Hajime_Saito
    Sep 8, 2021 at 2:25
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    $\begingroup$ Yeah, $\pi_* \mathcal O_Y = \mathcal O_X$ implies $\pi_* \mathbb G_m = \mathbb G_m$. I think $\pi_* \mathcal O_Y = \mathcal O_X$ follows from the fact that $H^0( SL_n/P, \mathcal O_{SL_n/P})$ is equal to the base field after applying flat base change and base-changing by the map from $X$ to the base field. $\endgroup$
    – Will Sawin
    Sep 8, 2021 at 13:09
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    $\begingroup$ The sheaf $R^1 \pi_* \mathbb G_m$ should just be $\mathbb Z^r$ (so constant, not just locally constant), for $r$ the rank of the center of the Levi of $P$, which is equivalent to the statement that, for arbitrary $X$, a line bundle on $Y$ has a $\mathbb Z^r$-valued invariant, the multi-degree of its restriction to $SL_n/P$, and two line bundles with the same degree are equal up to tensoring with the pullback of a line bundle from $X$. This follows from the same kind of argument, noting that the pushforward of a line bundle trivial on the fiber is a line bundle on $X$. $\endgroup$
    – Will Sawin
    Sep 8, 2021 at 13:09

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