smooth manifolds and affine varieties

[ForMyThunder]

This is really just a general question of interest: can every smooth n-manifold be embedded (in R2n+1 say) so that it coincides with an affine variety over R? Does anyone know of any results on this?

[CompuChip]

You can do even better than 2n + 1... this is a (rather deep) theorem (http://en.wikipedia.org/wiki/Nash_embedding_theorem).

[quasar987]

My ignorance is complete when it comes to algebraic geometry but I read on wikipedia that unless the field is finite, the zariski topology is never hausdorff. So since R is not finite and embedded manifolds are hausdorff, no affine variety can be identitfied topologically with a submanifold of R^N... Ok, so I guess you're asking if every manifold can be embedded in R^N so that it coincides as sets with an affine variety.

[micromass]

ForMyThunder: you'll be very interested in GAGA-style results. These theorems try to associate a manifold (actually something more general) to an affine variety. This GAGA-correspondence is very nice because Hausdorff spaces correspond to separated varieties, compact spaces correspond to complete (proper) varieties, etc.

I suggest you read the excellent book "Algebraic and analytic geometry" by Neeman. I think this is exactly what you want!!

[mathwonk]

look up theorems and conjectures of nash and kolla'r.

e.g.:

http://books.google.com/books?id=AmYklA6B9hYC&pg=PA137&lpg=PA137&dq=conjectures+of+nash,+theorems+of+kollar&source=bl&ots=rugYy1eIeU&sig=4HOn3EaEs8XCE4g_ly6Ggbbxz4A&hl=en#v=onepage&q&f=false