Automorphisms of the affine Cremona group

We show that every automorphism of the group Gn := Aut(An) of polynomial automorphisms of complex affine n-space An = Cn is inner up to field automorphisms when restricted to the subgroup TGn of tame automorphisms. This generalizes a result of Julie Deserti who proved this in dimension n = 2 where all automorphisms are tame: TG2 = G2. The methods are different, based on arguments from algebraic group actions

On Maximal Subalgebras

Let $\textbf{k}$ be an algebraically closed field. We classify all maximal $\textbf{k}$-subalgebras of any one-dimensional finitely generated $\textbf{k}$-domain. In dimension two, we classify all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, t^{-1}, y]$. To the authors' knowledge, this is the first such classification result for an algebra of dimension $> 1$. In the course of this study, we classify also all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that contain a coordinate. Furthermore, we give examples of maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that do not contain a coordinate.Comment: 30 pages, typos corrected, minor changes, improved expositio

Holomorphically Equivalent Algebraic Embeddings

We prove that two algebraic embeddings of a smooth variety $X$ in $\mathbb{C}^m$ are the same up to a holomorphic coordinate change, provided that $2 \dim X + 1$ is smaller than or equal to $m$. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of $\mathbb{C}^m$.Comment: 17 pages. Version 2 acknowledges the fact that the main result of this paper was previously established by Kaliman, see http://arxiv.org/abs/1309.379

Automorphisms of C3\mathbb{C}^3 Commuting with a C+\mathbb{C}^+-Action

Let $\rho$ be an algebraic action of the additive group $\mathbb{C}^+$ on the three-dimensional affine space $\mathbb{C}^3$. We describe the group $\textrm{Cent}(\rho)$ of polynomial automorphisms of $\mathbb{C}^3$ that commute with $\rho$. A particular emphasis lies in the description of the automorphisms in $\textrm{Cent}(\rho)$ coming from algebraic $\mathbb{C}^+$-actions. As an application we prove that the automorphisms in $\textrm{Cent}(\rho)$ that are the identity on the algebraic quotient of $\rho$ form a characteristic subgroup of $\textrm{Cent}(\rho)$.Comment: 18 pages, typos corrected, minor changes, improved expositio

Automorphisms of the plane preserving a curve

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.Comment: 21 pages, typos corrected, minor changes, improved expositio

On the Topologies on ind-Varieties and related Irreducibility Questions

In the literature there are two ways of endowing an affine ind-variety 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 of Shafarevich which is different from a counter-example given by Homma. We finish the paper with an irreducibility criterion similar to the one given by Shafarevich.Comment: 11 pages, typos corrected, minor changes, improved expositio

Contributions to automorphisms of affine spaces

We study aspects of the group G_n of polynomial automorphisms of the affine space A^n, the so-called affine Cremona group. Shafarevich introduced on G_n the structure of an ind-variety, an infinite-dimensional analogon to a (classical) variety. The aim of this thesis is to study G_n within the framework of ind-varieties. The thesis consists of five articles. In the following we summarize them. 1. On the Topologies on ind-Varieties and related Irreducibility Questions. In the literature there are two ways of endowing an affine ind-variety with a topology. One possibility is due to Shafarevich and the other due to Kambayashi. 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 by Shafarevich which is different from a counter-example given by Homma. We finish the article with an irreducibility criterion similar to the one given by Shafarevich. 2. On Automorphisms of the Affine Cremona Group (joint with Hanspeter Kraft) We show that every automorphism of the group G_n is inner up to field automorphisms when restricted to the subgroup TG_n of tame automorphisms. This generalizes a result of Julie Déserti who proved this in dimension n = 2 where all automorphisms are tame, i.e. TG_2 = G_2. The methods are different, based on arguments from algebraic group actions. 3. A Note on Automorphisms of the Affine Cremona Group Let G be an ind-group and let U be a unipotent ind-subgroup. We prove that an abstract automorphism f: G -> G maps U isomorphically onto a unipotent ind-subgroup of G, provided that f fixes a closed torus T in G that normalizes U and the action of T on U by conjugation fixes only the neutral element. As an application we generalize the main result of the article "On Automorphisms of the Affine Cremona Group" as follows: If an abstract automorphism of G_3 fixes the subgroup of tame automorphisms TG_3, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism). 4. Automorphisms of the Plane Preserving a Curve (joint with Jérémy Blanc) 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. 5. Centralizer of a Unipotent Automorphism in the Affine Cremona Group Let g be a unipotent element of G_3. We describe the centralizer Cent(g) inside G_3. First, we treat the case when g is a modified translation. In the other case, we describe the subset Cent(g)_u of unipotent elements of Cent(g) and prove that it is a closed normal subgroup of Cent(g). Moreover, we show that Cent(g) is the semi-direct product of Cent(g)_u with a closed algebraic subgroup R of Cent(g). Finally, we prove that the subgroup of Cent(g) consisting of those elements that induce the identity on the algebraic quotient Spec O(A^3)^g form a characteristic subgroup of Cent(g)

A note on Automorphisms of the Affine Cremona Group

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 G3 in dimension 3 fixes the subgroup of tame automorphisms TG3, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism)

Algebraic embeddings of ℂ into SL_n(ℂ)

We prove that any two algebraic embeddings ℂ → SL n (ℂ) are the same up to an algebraic automorphism of SL n (ℂ), provided that n is at least 3. Moreover, we prove that two algebraic embeddings ℂ → SL2(ℂ) are the same up to a holomorphic automorphism of SL2(ℂ)