# ag.algebraic geometry – Higher direct images of \$mathbb{G}_m\$ under a projective bundle

Let $$X$$ be a smooth projective rational variety over $$mathbb{C}$$, and let $$pi:Yrightarrow X$$ 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 $$R^ipi_*(mathbb{G}_m)$$ on $$X$$? For example, is $$pi_*mathbb{G}_m$$ a locally constant sheaf on $$X$$?

Any help would be appreciated.