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^*)$which is forgetting the flat connection. 1 I think the right thing to look at is$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.