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Is affine $n$-space determined by its automorphism group?

Le : 18/05/2018 14h00
Par : Hanspeter Kraft (Bâle)
Lieu : i 001
Lien web :
Résumé : If two topological manifolds have isomorphic automorphism groups, then they are isomorphic, and the same is true for differentiable manifolds. These results go back to the Whittaker (1963) and Filipkiewicz (1982), and the question arises if similar results hold in the algebraic setting. Of course, we have to make some additional assumptions, since there are many varieties with trivial automorphism group. Our main example is affine $n$-space $A^n$, $n > 1$. In this case, $Aut(A^n)$ is a so-called ind-group, i.e. an infinite dimensional algebraic group, a notion introduced in 1966 by Shafarevich. We have shown earlier that for an affine variety $X$ the existence of a group isomorphism $f : Aut(A^n) \to Aut(X)$ which maps algebraic subgroups to algebraic subgroups implies that $X$ is isomorphic to $A^n$. In our present work we start with an arbitrary isomorphism $f$, but we make additional assumptions on $X$. Theorem: Assume that $X$ is a quasi-projective $n$-dimensional variety, $n>1$, and that $Aut(X)$ is isomorphic to $\Aut(A^n)$. Then $X$ is isomorphic to $A^n$ if one of the following two conditions is satisfied: (a) $X$ is smooth, $Pic(X)$ is finite, and the Euler-characteristic of $X$ is nonzero. (b) $X$ is quasi-affine and toric. In the talk, we will discuss automorphism groups of varieties, give s few examples and then shortly indicate some proofs.