Let $G$ be a group acting properly by biholomorphisms on a complex manifold $X$, so $X // G$ is a complex orbifold. Let the *holomorphic Picard group* $Pic\_{hol}(X//G)$ be the group of isomorphism classes of $G$-equivariant holomorphic line bundles on $X$, under tensor product. This is naturally isomorphic to the group $H^1(X/G; \mathcal{O}^\times)$.


The first Chern class furnishes a map
$$c\_1 : Pic\_{hol}(X//G) \to H^2(X//G;\mathbb{Z})$$
to the second integral cohomology of the obifold (where cohomology is taken in the orbifold sense: it is not the integral cohomology of the actual quotient $X/G$). This is the connecting homomorphism for the exponential sequence $\mathbb{Z} \to \mathcal{O} \to \mathcal{O}^\times$ of sheaves on $X//G$.


I am interested in conditions on the orbifold so that this map is *injective*, and am happy to suppose that $H^1(X//G;\mathbb{Z})=0$. (Please do not tell me that the condition I want is that $H^1(X//G;\mathcal{O})=0$: I know this, and want conditions for it to hold.)



Picard group of complex orbifolds
I guess that for reductive $G$ one has $H^1(X//G,O) = H^1(X,O)^G$ (the $G$-invariants). So, the condition you want is $H^1(X,O)^G = 0$. For nonreductive $G$ there is a spectral sequence $H^q(G,H^p(X,O)) \Rightarrow H^{p+q}(X//G,O)$ so the condition is
$$
H^1(G,H^0(X,O)) = 0,
\qquad
Ker(H^1(X,O)^G \to H^2(G,H^0(X,O))) = 0.
$$


