Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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?
     
  2. jcsd
  3. Sep 26, 2011 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

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

    quasar987

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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

    mathwonk

    User Avatar
    Science Advisor
    Homework Helper

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Smooth manifolds and affine varieties
Loading...