Picard groups of certain compact complex parallelizable manifolds and related spaces

Abstract

Let $G$ be a complex simply connected semisimple Lie group and let $\Gamma$ be a torsionless uniform irreducible lattice in $G$. Then $\Gamma\backslash G$ is a compact complex non-K\"ahler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of $\Gamma\backslash G$ when $\rank(G)\geq 3$. When $\rank(G)\lneq 3$, we determine the group $Pic^0(\Gamma\backslash G)\subset Pic(\Gamma\backslash G)$ of topologically trivial holomorphic line bundles. When $\rank(G)\ge 2$, we also show that $Pic^0(P_\Gamma)$ is isomorphic to $Pic^0(Y)$ where $P_\Gamma$ is a $\Gamma\backslash G$-bundle associated to a principal $G$-bundle over a compact connected complex manifold $Y$, and, when $\rank(G)\ge 3$, we show that $Pic(Y)\to Pic(P_\Gamma)$ is injective with finite cokernel.