= Holomorphic Picard group
{title2=$\operatorname{Pic}_{\rm hol}(X)=H^1(X,\mathcal O_X^*)$}
= Analytic Picard group
{synonym}
The <holomorphic Picard group> consists of isomorphism classes of <holomorphic line bundles>, with <tensor product> as its group law. The trivial bundle is the identity and dualization gives inverses. Transition functions identify it with $H^1(X,\mathcal O_X^*)$. On the <complex projective line> every class is $\mathcal O(n)$, so degree gives an isomorphism with $\mathbb Z$.
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