Is there a generalization of Borel-Weil-Bott for partial flag varieties, i.e. homogeneous spaces of the form $G/P$ with $P$ parabolic and $G$ semisimple? If so, I would like a reference.
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7$\begingroup$ Doesn't this follow from the usual Borel-Weil-Bott by noting that the (derived) pushforward from $G/B$ to $G/P$ of a line bundle pulled back from $G/P$ is again that line bundle? $\endgroup$– Will SawinCommented Nov 9, 2020 at 21:54
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$\begingroup$ @WillSawin How does that help? $\endgroup$– Avi SteinerCommented Nov 9, 2020 at 22:01
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2$\begingroup$ The cohomology of any equivariant line bundle on $G/P$ is equal to the cohomology of its pullback to $G/B$, which can be calculated using the usual Borel-Weil-Bott, by the Leray spectral sequence. $\endgroup$– Will SawinCommented Nov 9, 2020 at 22:10
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1$\begingroup$ Section 2.3 of arxiv.org/abs/1203.2575 and references therein should be useful. $\endgroup$– JefCommented Nov 9, 2020 at 22:48
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3$\begingroup$ @AviSteiner Because the fibers are connected (and have no coherent cohomology, if you're interested in the derived pushforward). This is a local question, so we can work locally on $G/P$, and in particular assume the line bundle is trivial - it's a question about $f_* \mathcal O_{G/B}$. We know this is trivial for a proper separable map with reduced connected fibers. $\endgroup$– Will SawinCommented Nov 10, 2020 at 12:14
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1 Answer
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There are many places that give a complete answer to your question: One is a paper in the Annals of Math written by Kostant around the middle fifties, other more geometrical is due to Griffits-Schmid published in Acta Mathematica in the late sixties. best regards
