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.)