Embeddings of smooth affine varieties into linear algebraic groups
This is joint work with Peter Feller. There are the following fundamental questions concerning embeddings from an object Z into another object X:
1. (Existence) Does there exists an embedding of Z into X?
2. (Uniqueness) Having two embeddings f, g of Z into X, does there exists an automorphism ψ of X such that g = ψ ◦ f?
In this talk, we will mainly focus on the first question in the category of affine varieties, where Z is smooth and X is a
linear algebraic group. Amongst other things, we will discuss the following result.
Theorem. For every simple linear algebraic group G and every smooth affine variety Z with dim G > 2 dim Z + 1, there exists an embedding of Z into G.
The proof is based upon parametric transversality results for flexible affine varieties due to Kaliman. We will also discuss the following result.
Proposition. For every non-finite algebraic group G and every d ≥ dim G / 2, there exists an irreducible smooth affine variety of dimension d that does not admit an embedding into G.
This proposition implies the optimality of the above existence theorem up to a possible improvement of the dimension bound by one. It’s proof is an adaptation of a Chow-group-based argument due to Bloch, Murthy, and Szpiro for the affine space.
- Informed public
- Christian Urech
- Monique Kiener (if you want to attend to the seminar by zoom, please contact me, and I'll give you the link)