Homological methods in certain Picard group computations

Abstract

Let G be a connected complex semisimple Lie group, Γ be a cocompact, irreducible and torsionless lattice in G and K be a maximal compact subgroup of G. Assume Γ acts by left multiplication and K acts by right multiplication on G. Let MΓ=Γ﹨G, X=G/K and XΓ=Γ﹨X. In this article we prove that for any n≥0, the composition Hn(XΓ,C)→Hn(MΓ,C)→Hn(MΓ,OMΓ) is an isomorphism. As an application when G is simply connected, we compute the Picard group of MΓ for the cases rank(G) =1,2. More precisely we show that if rank(G) =1, Pic(MΓ)=(Cr/Zr)⊕A and if rank(G) =2, then Pic(MΓ)≅A via the first Chern class map, where A is the torsion subgroup of H2(MΓ,Z) and r is the rank of Γ/[Γ,Γ].