Seminar Algebra and Geometry: Hanspeter Kraft (Universität Basel)

Affine n-Space is Characterized by its Automorphisms

We show that the automorphism group of affinen-space Aⁿ determines Aⁿ up to isomorphism: If X is a connected affine variety such that Aut(X) \simeq Aut(Aⁿ) as ind-groups, then X \simeq Aⁿ as varieties.
We also show that every torus appears as Aut(X) for a suitable affine variety X, but that Aut(X) cannot be isomorphic to a semisimple group. In fact, if Aut(X) is finitedimensional and if X \not\simeq A¹, then the connected component Aut(X)^◦ is a torus.
Concerning the structure of Aut(Aⁿ) we prove that any homomorphism Aut(An)→ G of ind-groups either factors through jac : Aut(Aⁿ)→C^∗ where jac is the Jacobian determinant, or it is a closed immersion. For SAut(Aⁿ) := ker(jac)⊂Aut(Aⁿ) we show that everynontrivial homomorphism SAut(An)→G is a closed immersion.
Finally, we prove that every non-trivial homomorphism φ: SAut(Aⁿ)→SAut(Aⁿ) is anautomorphism, and that φ is given by conjugation with an element from Aut(Aⁿ).

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