GAGA

Remarks

The idea of the proof: both statements are first checked on $\mathbb{P}^n$, by comparing the cohomology of the sheaves $\mathcal{O}(d)$, which is the same in both settings. For a general coherent sheaf one uses that it has a resolution by sums of sheaves $\mathcal{O}(d)$, together with Serre's theorems on twisting and induction on dimension.

Example: by part 2 applied to ideal sheaves, every closed analytic subset of $\mathbb{P}^n_\C$ is algebraic; this is Chow's theorem. In particular, every compact complex manifold that embeds holomorphically in $\mathbb{P}^n_\C$ is a projective algebraic variety, and every holomorphic map between projective varieties is algebraic.

The properness of $X$ is essential. On $X = \mathbb{A}^1_\C$, the holomorphic function $e^z$ is a global section of $\mathcal{O}_{X^{an}}$ that is not a polynomial. So part 1 fails already for $i = 0$ and $\mathcal{F} = \mathcal{O}_X$. The theorem extends to all proper schemes over $\C$, by a result of Grothendieck.

Going deeper

For a student

GAGA is what allows results proved with analytic tools, such as Hodge theory and the Kodaira vanishing theorem, to be stated and used for algebraic varieties. It also shows that the Picard group of a projective variety equals the group of holomorphic line bundles on the associated complex manifold. For non-proper varieties the two theories differ, as the example of $e^z$ on the affine line shows.