Smooth manifolds and affine varieties

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Discussion Overview

The discussion revolves around the question of whether every smooth n-manifold can be embedded in R2n+1 such that it coincides with an affine variety over R. Participants explore connections between smooth manifolds and affine varieties, touching on concepts from algebraic geometry and topology.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether every smooth n-manifold can be embedded in R2n+1 to coincide with an affine variety over R, seeking results on this topic.
  • Another participant suggests that a deeper theorem allows for embeddings in dimensions better than R2n+1.
  • A different viewpoint highlights a potential issue with the Zariski topology, noting that it is never Hausdorff unless the field is finite, which raises questions about the topological identification of affine varieties with submanifolds of RN.
  • One participant introduces GAGA-style results, which relate manifolds to affine varieties, emphasizing the correspondence between Hausdorff spaces and separated varieties, as well as compact spaces and complete varieties.
  • A suggestion is made to look into theorems and conjectures by Nash and Kollár for further insights on the topic.

Areas of Agreement / Disagreement

Participants express differing views on the embedding of smooth manifolds in relation to affine varieties, with no consensus reached on the initial question. Some participants propose connections and theorems that may be relevant, while others raise concerns about topological implications.

Contextual Notes

The discussion includes unresolved mathematical considerations regarding the properties of the Zariski topology and the implications for embedding manifolds as affine varieties. The relationship between Hausdorff spaces and affine varieties is also noted as a point of contention.

Who May Find This Useful

This discussion may be of interest to those studying algebraic geometry, topology, and the relationships between different mathematical structures, particularly in the context of smooth manifolds and affine varieties.

ForMyThunder
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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!
 

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