What is the role of projective spaces in GAGA?

Question

The GAGA theorem is a celebrated elaboration of the idea that complex analytic and complex algebraic geometry are equivalent, at least for smooth projective varieties/manifolds.

I am aware why this is a theorem about projective varieties; historically the two classes of varieties people cared about were projective and affine varieties. I am also aware that GAGA fails to hold for affine varieties and Stein manifolds.

I wonder if there is any deeper/conceptual reason why projective spaces in particular ought to appear though?

For instance I feel like I understand why projective spaces appear when you work with bundles, say when working with Chern classes since they classify complex vector bundles. But GAGA is a theorem about categories of coherent sheaves and not just vector bundles ...

Maybe if one does "relative" GAGA in the sense of Grothendieck, can we work over bases other than projective spaces or more generally projective varieties? And even if we can, is there some significance in choosing projective spaces anyway?

Thanks a lot and sorry if the question is too vague and philosophical!

Answer 1

The Serre comparison theorems are valid for complete (= proper) varieties over ${\Bbb C}$, with no relation to projective space. See this talk by Grothendieck (Séminaire Cartan 9 (1956-1957), Exposé No. 2).

Answer 2

Short version of the proof of GAGA is this: it's an immediate consequence of Serre's twisting theorem that every analytic coherent sheaf $\mathcal{F}$ is a quotient of $\mathcal{O}^{\mathrm{an}}(-m)^{\oplus n}$ by the image of a map from $\mathcal{O}^{\mathrm{an}}(-m')^{\oplus n'}$. That map must be algebraic (that's where properness is important!); in fact it's just a $n\times n'$ matrix of polynomials of degree $m'-m$. So we can get the desired algebraic coherent sheaf we want by taking the cokernel of the "same" map $\mathcal{O}(-m')^{\oplus n'}\to \mathcal{O}(-m)^{\oplus n}$.

Morally, I would say the important thing about properness is that analytifying is fully faithful. The essence of the proof is that once you know this, you just need a set of algebraic coherent sheaves whose analytifications generate the derived category of analytic coherent sheaves and you'll be done. In the projective case, this is easy: you just use $\mathcal{O}(-m)$; in the general proper case, this is harder, but possible; that's what Grothendieck is doing on the last page of the article cited above.