Smooth manifolds and affine varieties

In summary, the conversation discusses the possibility of every smooth n-manifold being embedded in R^N as an affine variety, and references GAGA-style results and theorems/conjectures of Nash and Kollár as resources for further exploration.
  • #1
ForMyThunder
149
0
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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  • #2
You can do even better than 2n + 1... this is a (rather deep) theorem.
 
  • #3
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.
 
  • #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!
 

What is a smooth manifold?

A smooth manifold is a mathematical object that is locally like Euclidean space. It is a topological space that is locally homeomorphic to a vector space, meaning that it is locally flat and smooth. In other words, a smooth manifold is a space that is smooth and continuous, but not necessarily straight.

What is an affine variety?

An affine variety is a set of solutions to a system of polynomial equations. It is a geometric object that is defined by a set of polynomial equations and can be represented as a subset of n-dimensional space. Affine varieties are important in algebraic geometry and have many applications in mathematics and science.

What is the relationship between smooth manifolds and affine varieties?

Smooth manifolds and affine varieties are closely related. In fact, every smooth manifold can be viewed as an affine variety, and every affine variety can be viewed as a smooth manifold. This relationship is known as the Nash embedding theorem, which states that every smooth manifold can be embedded into a higher-dimensional Euclidean space.

What are some applications of smooth manifolds and affine varieties?

Smooth manifolds and affine varieties have many applications in various fields of mathematics and science. They are used in algebraic geometry, differential geometry, topology, physics, and more. They are particularly useful in studying the geometry and dynamics of complex systems, such as in chaos theory and control theory.

What is the difference between smooth manifolds and topological manifolds?

The main difference between smooth manifolds and topological manifolds is that smooth manifolds have additional structure, namely a smooth structure. This means that smooth manifolds have a well-defined notion of differentiation and are equipped with smooth maps between them. On the other hand, topological manifolds only have a topological structure and do not have a concept of smoothness.

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