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Smooth manifolds and affine varieties

  1. Sep 26, 2011 #1
    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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  3. Sep 26, 2011 #2


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    You can do even better than 2n + 1... this is a (rather deep) theorem.
  4. Sep 26, 2011 #3


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    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.
  5. Sep 26, 2011 #4
    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!!
  6. Sep 28, 2011 #5


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