Seminar Algebra and Geometry: Immanuel van Santen (Basel)

Characterisation of varieties by their automorphisms

This is joint work with Hanspeter Kraft (University of Basel) and Andriy Regeta (University of Cologne). The main problem we address in this talk is the characterization of the affine space An by its automorphism group Aut(Aⁿ). More precisely, we ask, whether the existence of an abstract group isomorphism Aut(X) \simeq Aut(Aⁿ) implies the existence of an isomorphism of algebraic varieties X \simeq Aⁿ. The following is our main result. Main Theorem.
Let X be a quasi-affine irreducible variety such that Aut(X) \simeq Aut(Aⁿ). Then X \simeq Aⁿ if one of the following conditions holds.
(1) X is a Q-acyclic open subset of a smooth affine rational variety, and dim(X) is a most equal to n;
(2) X is toric and dim(X) is at least equal to n. After giving a brief history on some related results that concern the characterisation of geometric objects via their automorphisms, we give the key ideas of the proof of our main result.

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