In the literature there are two ways of endowing an affine indvariety with a topology. One possibility is due to Shafarevich and the other to Kambayashi. In this paper we specify a large class of affine ind-varieties where these two topologies differ. We give an example of an affine ind-variety that is reducible with respect to Shafarevich's topology, but irreducible with respect to Kambayashi's topology. Moreover, we give a counter-example of a supposed irreducibility criterion given in [Sha81] which is different from the counter-example given by Homma in [Kam96]. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich.
Abstract. We show that every automorphism of the group Gn := Aut(A n ) of polynomial automorphisms of complex affine n-space A n = C n is inner up to field automorphisms when restricted to the subgroup T Gn of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n = 2 where all automorphisms are tame: T G 2 = G 2 . The methods are different, based on arguments from algebraic groups actions.
Abstract. We prove that any two algebraic embeddings C Ñ SLnpCq are the same up to an algebraic automorphism of SLnpCq, provided that n is at least 3. Moreover, we prove that two algebraic embeddings C Ñ SL2pCq are the same up to a holomorphic automorphism of SL2pCq.
Abstract. We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the groups of positive dimension occuring is also given in the case where the curve is geometrically irreducible and the field is perfect.
Abstract. Let G be an ind-group and let U ⊆ G be a unipotent ind-subgroup. We prove that an abstract group automorphism θ : G → G maps U isomorphically onto a unipotent ind-subgroup of G, provided that θ fixes a closed torus T ⊆ G, which normalizes U and the action of T on U by conjugation fixes only the neutral element. As an application we generalize a result by Hanspeter Kraft and the author as follows: If an abstract group automorphism of the affine Cremona group G 3 in dimension 3 fixes the subgroup of tame automorphisms T G 3 , then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).
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