Picard groups of certain compact complex parallelizable manifolds and related spaces

TL;DR

This paper computes the Picard groups of certain compact complex manifolds formed from semisimple Lie groups and lattices, revealing their structure and relations to associated bundles and base manifolds.

Contribution

It provides explicit calculations of Picard groups for these manifolds, especially when the rank of the Lie group is at least 3, and explores their relation to associated bundles and base spaces.

Findings

Computed Picard groups for rank ≥ 3 cases.

Determined the structure of topologically trivial line bundles for lower ranks.

Established isomorphisms and injectivity relations between Picard groups of bundles and base manifolds.

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.