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I would like to know how to prove that : $ \mathrm{Pic} ( X ) \simeq H^1 ( X , \mathcal{O}_{X}^* ) $. I specially want to know how to prove that $ \mathrm{Pic} ( X ) \to H^1 ( X , \mathcal{O}_{X}^* ) $ is injective. $ X $ is a complex manifold. Thanks a lot.

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Check out this question. What you want is the rank one holomorphic version. – Michael Albanese Aug 29 '13 at 13:36
See also…. – John M Aug 29 '13 at 13:40
Besides checking other questions, you might also want to provide your definition of the Picard group. It's not uncommon to define it as the sheaf cohomology group, in which case there would be nothing to prove.. – Marek Aug 29 '13 at 14:04
@Bryan You should also provide the definition of $H^1$: for Čech cohomology this is fairly straightforward, but there is some extra work to be done for derived functor cohomology. – Zhen Lin Aug 29 '13 at 23:25

I think the best reference for that is the book by Griffiths and Harris "Principles of algebraic geometry", the chapter about divisors and line bundles.

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