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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.)

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up vote 3 down vote accepted

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 $$ H^1(G,H^0(X,O)) = 0, \qquad Ker(H^1(X,O)^G \to H^2(G,H^0(X,O))) = 0. $$

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Could you suggest a reference for this? –  Oscar Randal-Williams Sep 17 '10 at 8:48
I don't know a reference, the definition of the quotient stack implies that $\Gamma(X//G,F) = \Gamma(X,F)^G$ (where $F$ is considered as a $G$-equivariant sheaf on $X$). Now the spectral sequence of the composition of functors computes the higher cohomology. –  Sasha Sep 17 '10 at 10:48

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