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 $n\ge 0$, the composition $H^n(X_{\Gamma },\mathbb{C})\to H^n(M_{\Gamma },\mathbb{C})\to H^n(M_{\Gamma },{\mathcal{O}}_{M_{\Gamma }})$ 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 $\Gamma /[\Gamma ,\Gamma ]$.

arXiv comment10 pages