Smooth manifolds and affine varieties

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The discussion centers on whether every smooth n-manifold can be embedded in R^(2n+1) to coincide with an affine variety over R. It highlights a significant theorem regarding the embedding of manifolds and the implications of the Zariski topology, noting that since R is not finite, embedded manifolds are Hausdorff, preventing identification with affine varieties as submanifolds of R^N. The conversation references GAGA-style results, which connect manifolds to affine varieties, emphasizing the correspondence between Hausdorff and separated varieties, as well as compact and complete varieties. A recommendation is made for the book "Algebraic and Analytic Geometry" by Neeman for further exploration of these concepts. The discussion concludes with a suggestion to investigate theorems and conjectures by Nash and Kollár for deeper insights.
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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?
 
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You can do even better than 2n + 1... this is a (rather deep) theorem.
 
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.
 
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!
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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