Chow's Lemma states that every algebraic variety is birationally equivalent to a projective variety, and Hironaka's work on resolutions of singularities implies every projective variety is birationally equivalent to a non-singular projective variety. I am wondering whether every non-singular projective variety is then birationally equivalent to some complex projective manifold, i.e. a closed complex submanifold of $\mathbb{CP}^n$?
I am a differential geometer rather than algebraic, so forgive me if this is a well-known principle. On searching I have come across things such as GAGA and Chow's Theorem but I'm having a hard time figuring out if these are what I need – I can only seem to derive statements along the lines of 'to every non-singular projective variety we can associate a complex manifold'. But I don't know what 'associate' means here. Thanks in advance.
Best Answer
The answer is essentially yes: since every variety is birational to a projective variety and every projective variety is birational to a smooth projective variety, every variety is birational to a smooth projective variety. And from every smooth projective variety you can get a complex manifold in a canonical way. But the variety and the complex manifold won't be literally birational because of the difference between varieties and manifolds (generic points, eg). But if you're looking for the incarnation of a variety in the world of complex manifolds, the analytification is the thing.
Some references for analytification are https://ncatlab.org/nlab/show/analytification, and also https://www.math.ucdavis.edu/~osserman/classes/248B-W10/analytic-top.pdf , where it's spoken of as putting the analytic topology on a scheme.
One important point to remember here is that there are multiple different definitions of what a "variety" is. Some people believe that varieties over a field k are certain closed subsets of $k^n$ or $\Bbb P^n_k$ with the Zariski topology, other people believe that they're schemes of finite type over $k$. If you believe the first definition, smooth varieties over $\Bbb C$ are really complex manifolds. If you believe the second definition, to each smooth variety over $\Bbb C$, we can canonically associate a complex manifold. The latter viewpoint on what a variety is has been very useful over the past 60-70 years, and there's a canonical way to get from the first to the second in some cases, like varieties in the first sense over $\Bbb C$ with certain adjectives attached. And thinking about these varieties as in the first viewpoint is often a useful reasoning tool, but there are times where you'll have to be careful and explicit about what you're assuming. Especially when talking to people not already well-versed in the field. (This sort of confusion is a big problem for many learners - I went through it my self, for instance.)