(Serre's GAGA)
Let \(X\) be a projective scheme over \(\CC\) and \(X^{an}\) its associated complex analytic space, with the morphism of ringed spaces \(h \colon X^{an} \to X\).
For every coherent sheaf \(\mathcal{F}\) on \(X\), the natural maps \(H^i(X, \mathcal{F}) \to H^i(X^{an}, \mathcal{F}^{an})\) are isomorphisms, where \(\mathcal{F}^{an} = h^* \mathcal{F}\).
The functor \(\mathcal{F} \mapsto \mathcal{F}^{an}\) is an equivalence from coherent \(\OO_X\)-modules to coherent analytic sheaves on \(X^{an}\).
(Chow's theorem)
A closed analytic subspace of \(\PP^n(\CC)\) is algebraic, and a holomorphic map between projective varieties is a morphism.
Holomorphic line bundles on a projective variety are algebraic, so \(\Pic(X) \cong H^1(X^{an}, \OO^*_{X^{an}})\). Global meromorphic functions on a smooth projective variety are rational functions. Projectivity cannot be dropped: on \(X = \AA^1_\CC\), \(H^0(X, \OO_X) = \CC[z]\) while \(H^0(X^{an}, \OO_{X^{an}})\) is the ring of entire functions, and \(\exp \colon \CC \to \CC^*\) is holomorphic but not a morphism \(\AA^1 \to \AA^1 \setminus \{0\}\).