Picard groups of certain compact complex parallelizable manifolds and related spaces

Abstract

Let G be a complex simply connected semisimple Lie group and let Γ be a torsionless uniform irreducible lattice in G. Then Γ﹨G is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of Γ﹨G when rank(G)≥3. When rank(G)<3, we determine the group Pic0(Γ﹨G)⊂Pic(Γ﹨G) of topologically trivial holomorphic line bundles. When rank(G)≥2, we also show that Pic0(PΓ) is isomorphic to Pic0(Y) where PΓ is a Γ﹨G-bundle associated to a principal G-bundle over a compact connected complex manifold Y, and, when rank(G)≥3, we show that Pic(Y)→Pic(PΓ) is injective with finite cokernel.