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I think the right thing to look at is $H^1(X, \mathbb C^*)$. This classifies line bundles with a flat connection(there is no wikipedia link unfortunately), or equivalently, line bundles with locally constant transition functions. Now the natural embedding $\mathbb C C^* \to \mathbb O_X^$ O_X^*$ induces on a map on cohomology $H^1(X, \mathbb C^C^*) \to H^1(X, \mathbb O_X^*) O_X^*)$ which is forgetting the flat connection. |
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I think the right thing to look at is $H^1(X, \mathbb C^*)$. This classifies line bundles with a flat connection (there is no wikipedia link unfortunately), or equivalently, line bundles with locally constant transition functions. Now the natural embedding $\mathbb C \to \mathbb O_X^$ induces on a map on cohomology $H^1(X, \mathbb C^) \to H^1(X, \mathbb O_X^*) which is forgetting the flat connection. |
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