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If one has a quasiprojective complex variety X, there is a natural map from the algebraic Picard group to the analytic Picard group. Is this map either injective or surjective?

I assume the latter is not true, but couldn't think of an example (nor a definitive opinion for the former question) If you take a normal crossings projective compactification Xbar, then both notions coincide, but while an algebraic line bundle on Xbar that is trivial on X must be associated to the boundary divisor, I couldn't see if the same holds in the analytic setting.

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    Okay so a punctured higher genus curve shows that it isn't injective. Not sure about surjective.2011-03-15

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