Look at the following theorem:
The following two categories are equivalent:
- The category of non-singular projective varieties over $\mathbb C$. (Where a variety is understood as in Hartshorne's book i.e. a $\mathbb C$-scheme integral, of finite type and separated).
- The category of complex projective manifolds.
This powerful theorem should follow from the Serre's GAGA and from the Chow's theorem.
Now my question(s):
a) Is the statement of the above theorem, formally correct and precise?
b) Which are the "morphisms" of the two above categories? Morphisms of varieties for $1)$ and holomorphic maps for $2)$?
c) Is true that the theorem follows from GAGA+Chow?
Best Answer
The answer to the three questions is yes. If one wants to make it a little more precise, one could add the functor in play. This however isn't done in one line, which is why many people suppress it. The main ingredient of the proof is in fact Chow's theorem. Serre's GAGA is mainly concerned with the additional equivalence of coherent sheaves and cohomology etc. For the proof of the theorem, all GAGA is used for is that we want to recognise that the procedure is functorial. Without GAGA it roughly looks like this:
From Chow's theorem we get that every closed complex submanifold $X\subset \mathbb{P}^n$ is closed in the Zariski topology also and hence defines a complex projective variety in the sense of Hartshorne's first chapter (also with canonical embedding). It is not hard to see that the smooth projective varieties (with fixed embedding) in this way uniquely correspond to complex projective manifolds (with fixed embedding). It is easy to associates to a morphism of such embedded varieties a holomorphic map of the corresponding complex manifolds. Applying Chow's theorem again, this time to the graph of a holomorphic function between (embedded) complex projective manifolds, Segre embedded into some projective space, one gets that each holomorphic function is obtained from an algebraic one, so there is, if you want to tell it like this, an equivalence of the category of embedded varieties with the category of embedded manifolds.
To get rig of the additional datum of embeddings one could say by GAGA, ..., or one'd have to prove by hand that up to isomorphism, what we did above doesn't depend on the chosen embeddings.