Homological methods in certain Picard group computations

Abstract

Let $G$ be a connected complex semisimple Lie group, $\Gamma$ be a cocompact, irreducible and torsionless lattice in $G$ and $K$ be a maximal compact subgroup of $G$. Assume $\Gamma$ acts by left multiplication and $K$ acts by right multiplication on $G$. Let $M_{\Gamma}= \Gamma\backslash G$, $X=G/K$ and $X_{\Gamma}=\Gamma\backslash X$. In this article we prove that for any $n\geq0$, the composition $H^{n}(X_{\Gamma},\mathbb{C})\rightarrow H^{n}(M_{\Gamma},\mathbb{C})\rightarrow H^{n}(M_{\Gamma},\mathcal{O}_{M_{\Gamma}})$ is an isomorphism. As an application when $G$ is simply connected, we compute the Picard group of $M_{\Gamma}$ for the cases rank($G$) $=1,2$. More precisely we show that if rank($G$) $=1$, $Pic(M_{\Gamma})=(\mathbb{C}^{r}/\mathbb{Z}^{r})\oplus A$ and if rank($G$) $=2$, then $Pic(M_{\Gamma})\cong A$ via the first Chern class map, where $A$ is the torsion subgroup of $H^{2}(M_{\Gamma},\mathbb{Z})$ and $r$ is the rank of $\Gamma/[\Gamma,\Gamma]$.