Let $D$ be a bounded symmetric domain and $\Gamma\subset Aut(D)$ an arithmetic subgroup of automorphisms of $D$. Write $X=\Gamma\backslash D$, it is a connected Shimura variety. Suppose that $X$ is compact. I haven't found in the literature a discussion about the Picard rank of $X$ or about the singular cohomology of $X$ (for instance, what is $H^2(X,\mathbb{Z})$? What's the rank? Does it have torsion?). Does anyone know any reference for this type of questions?
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2$\begingroup$ This paper of A. Borel gives some information about real cohomology. $\endgroup$– abxCommented Nov 15, 2019 at 17:18
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$\begingroup$ Minor nitpick : if I remember correctly this locally symmetric variety is not necessarily a Shimura variety. The arithmetic group Gamma should also be congruence, right? $\endgroup$– Ariyan JavanpeykarCommented Nov 15, 2019 at 21:12
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$\begingroup$ Ariyan: Yes, you are correct, I was sloppy there. $\endgroup$– ChrisCommented Nov 15, 2019 at 23:30
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