Homological methods in certain Picard group computations

TL;DR

This paper uses homological methods to analyze Picard groups of certain quotients of complex semisimple Lie groups, establishing isomorphisms between cohomology groups and computing Picard groups for specific ranks.

Contribution

It proves an isomorphism between cohomology groups and computes the Picard group for quotients of complex semisimple Lie groups with rank 1 and 2.

Findings

The composition of cohomology maps is an isomorphism for all n.

For rank(G)=1, Picard group has a specific structure involving torsion and free parts.

For rank(G)=2, Picard group is isomorphic to the torsion subgroup of second integral cohomology.

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 $\Gamma/[\Gamma,\Gamma]$.