How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there any good references?
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3$\begingroup$ These varieties are iterated $P^1$-fibrations. This can be used to compute the cohomology. $\endgroup$– SashaCommented Jul 4, 2014 at 8:54
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$\begingroup$ @Sasha: I guess this is only the partial answer. There would also have to be a method to iteratively determine the first Chern class of the rank two bundle giving the $\mathbb{P}^1$-fibration. $\endgroup$– Matthias WendtCommented Jul 4, 2014 at 12:18
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4$\begingroup$ The additive structure is completely described by Sasha's comment - additively, the cohomology of the Bott-Samelson variety is the same as for $(\mathbb{P}^1)^{\times n}$. If you are additionally interested in the ring structure of generalized cohomology theories of Bott-Samelson varieties, you might want to check out part 2 of arXiv:0905.1341 of Calmès, Petrov and Zainoulline (as well as the references in there, to the classics of Demazure, Bott etc.). $\endgroup$– Matthias WendtCommented Jul 4, 2014 at 18:34
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2$\begingroup$ This is exactly what the original Bott and Samelson paper is about, and you should look at it. (Only later did people even realize that these manifolds are algebraic varieties, much less that they provide resolutions of singularities.) $\endgroup$– Allen KnutsonCommented Jul 8, 2014 at 3:42
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