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!
Best Answer
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).