arXiv subjects
Publications and source records attributed to Hanspeter Kraft.
Ramanujam's theorem states that any connected finite-dimensional subgroup of the automorphism group $\mathrm{Aut}(X)$ of an irreducible variety $X$ is an algebraic group, in a natural way. In this note, we discuss the notion of dimension and extend Ramanujam's theorem to arbitrary (not necessarily irreducible) varieties.
Let $X$ be a variety of dimension $n$, and let $\mathrm{Aut}(X)$ be its automorphism group. When $X$ is quasi-affine, we prove that a solvable subgroup of $\mathrm{Aut}(X)$ that is generated by an irreducible family of automorphisms containing the identity is an algebraic subgroup. Our main applications concern arbitrary varieties. First, every connected solvable subgroup of $\mathrm{Aut}(X)$ is contained in a Borel subgroup and its derived length is $\leq n+1$. Second, the notion of solvable and unipotent radicals are well defined for any subgroup of $\mathrm{Aut}(X)$. Third, if $X$ is quasi-affine and connected and $\mathcal{B} \subset \mathrm{Aut}(X)$ is a Borel subgroup of derived length $n+1$, then $X$ is isomorphic to the affine $n$-space $\mathbb{A}^n$ and $\mathcal{B}$ is conjugate to the Jonqui\`eres subgroup.
This is a brief overview of a few selected chapters on automorphism groups of affine varieties. It includes some open questions.
The automorphism group Aut(X) of an affine variety X is an ind-group. Its Lie algebra is canonically embedded into the Lie algebra VF(X) of vector fields on X. We study the relations between subgroups of Aut(X) and Lie subalgebras of VF(X). We show that a subgroup G of Aut(X) generated by a family of connected algebraic subgroups G_i of Aut(X) is algebraic if and only if the Lie algebras Lie G_i generate a finite dimensional Lie subalgebra of VF(X). Extending a result by Cohen-Draisma we prove that a locally finite Lie algebra L of VF(X) generated by locally nilpotent vector fields is algebraic, i.e. L = Lie G for an algebraic subgroup G of Aut(X). Along the same lines we prove that if a subgroup G of Aut(X) generated by finitely many connected algebraic groups is solvable, then it is a solvable algebraic group. We also show that the derived length a unipotent algebraic subgroup U of Aut(X) is bounded above by dim X. This result is based on the following triangulation theorem: Every unipotent algebraic subgroup of Aut(A^n) with a dense orbit in A^n is conjugate to a subgroup of the de Jonquières subgroup. Furthermore, we give an example of a free subgroup F of Aut(A^2) generated by two algebraic elements such that the Zariski closure of F is a free product of two nested commutative closed unipotent ind-subgroups. To any affine ind-group G one can associate a canonical ideal L_G \subset Lie G. It is linearly generated by the tangent spaces T_e X for all algebraic subsets X \subset G which are smooth in e. It has the important property that for a surjective homomorphism ϕ: G \to H the induced homomorphism dϕ_e : L_G \to L_H is surjective as well. Moreover, if H \subset G is a subnormal closed ind-subgroup of finite codimension, then L_H has finite codimension in L_G.
Let $k$ be an algebraically closed field of characteristic 0, and let $V$ be a finite-dimensional vector space. Let $End(V)$ be the semigroup of all polynomial endomorphisms of $V$. Let $E$ be a subset of $End(V)$ which is a linear subspace and also a semi-subgroup. Both $End(V)$ and $E$ are ind-varieties which act on $V$ in the obvious way. In this paper, we study important aspects of such actions. We assign to $E$ a linear subspace $D_{E}$ of the vector fields on $V$. A subvariety $X$ of $V$ is said to $D_{E}$ -invariant if $h(x)$ is in the tangent space of $x$ for all $h$ in $D_{E}$ and $x$ in $X$. We show that $X$ is $D_{E}$ -invariant if and only if it is the union of $E$-orbits. For such $X$, we define first integrals and construct a quotient space for the $E$-action. An important case occurs when $G$ is an algebraic subgroup of $GL(V$) and $E$ consists of the $G$-equivariant polynomial endomorphisms. In this case, the associated $D_{E}$ is the space the $G$-invariant vector fields. A significant question here is whether there are non-constant $G$-invariant first integrals on $X$. As examples, we study the adjoint representation, orbit closures of highest weight vectors, and representations of the additive group. We also look at finite-dimensional irreducible representations of SL2 and its nullcone.
An affine varieties with an action of a semisimple group $G$ is called "small" if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a highest weight vector. Such a variety $X$ carries a canonical action of the multiplicative group $\mathbb{K}^*$ commuting with the $G$-action. We show that $X$ is determined by the $\mathbb{K}^*$-variety $X^U$ of fixed points under a maximal unipotent subgroups $U$ of $G$. Moreover, if $X$ is smooth, then $X$ is a $G$-vector bundle over the quotient $X// G$. If $G$ is of type $A_n$ ($n>1$), $C_n$, $E_6$, $E_7$ or $E_8$, we show that all affine $G$-varieties up to a certain dimension are small. As a consequence we have the following result. If $n>4$, every smooth affine $SL_n$-variety of dimension $<2n$ is an $\mathrm{SL}_n$-vector bundle over the smooth quotient $X//\mathrm{SL}_n$, with fiber isomorphic to the natural representation or its dual.
We discuss the classical, and forgotten, notion of perpetuants. We give a proof of the Theorem of Stroh computing their dimensions, and exhibit a basis of perpetuants, thus closing an old line of investigation.
This article is a survey on ind-varieties and ind-groups introduced by Shafarevich in 1965, with a special emphasis on automorphism groups of affine varieties and actions of ind-groups on ind-varieties. We give precise definitions and complete proofs, including several known results. The survey contains many examples and also some questions which came up during our work on the subject. Among the new results we show that for an affine variety X the automorphism group Aut(X) is always locally closed in the ind-semigroup End(X) of all endomorphisms, and we give an example of a strict closed subgroup of a connected ind-group which has the same Lie algebra, based on the work of Shestakov-Umirbaev on the existence of non-tame automorphisms of affine 3-space.
We give a modern proof of the Regularization Theorem of André Weil which says that for every rational action of an algebraic group $G$ on a variety $X$ there exist a variety $Y$ with a regular action of $G$ and a $G$-equivariant birational map $X \to Y$. Moreover, we show that a rational action of $G$ on an affine variety $X$ with the property that each $g$ from a dense subgroup of $G$ induces a regular automorphism of $X$, is a regular action.
In this note we study the problem of characterizing the complex affine space $\mathbb{A}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\mathrm{Aut}(X)$ and $\mathrm{Aut}(\mathbb{A}^n)$ are isomorphic as abstract groups. If $X$ is either quasi-affine and toric or $X$ is smooth with Euler characteristic $χ(X) \neq 0$ and finite Picard group $\mathrm{Pic}(X)$, then $X$ is isomorphic to $\mathbb{A}^n$. The main ingredient is the following result. Let $X$ be a smooth irreducible quasi-projective variety of dimension $n$ with finite $\mathrm{Pic}(X)$. If $X$ admits a faithful $(\mathbb{Z} / p \mathbb{Z})^n$-action for a prime $p$ and $χ(X)$ is not divisible by $p$, then the identity component of the centralizer $\mathrm{Cent}_{\mathrm{Aut}(X)}( (\mathbb{Z} / p \mathbb{Z})^n)$ is a torus.
We show that every automorphism of the group $\mathcal{G}_n:= \textrm{Aut}(\mathbb{A}^n)$ of polynomial automorphisms of complex affine $n$-space $\mathbb{A}^n=\mathbb{C}^n$ is inner up to field automorphisms when restricted to the subgroup $T \mathcal{G}_n$ of tame automorphisms. This generalizes a result of \textsc{Julie Deserti} who proved this in dimension $n=2$ where all automorphisms are tame: $T \mathcal{G}_2 = \mathcal{G}_2$.
We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and 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 finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus. Concerning the structure of $Aut(A^n)$ we prove that any homomorphism $Aut(A^n) \to G$ of ind-groups either factors through the Jacobian determinant $jac\colon Aut(A^n) \to k^*$, or it is a closed immersion. For $SAut(A^n):=\ker(jac)$ we show that every nontrivial homomorphism $SAut(A^n) \to G$ is a closed immersion. Finally, we prove that every non-trivial homomorphism $SAut(A^n) \to SAut(A^n)$ is an automorphism, and is given by conjugation with an element from $Aut(A^n)$.
We show that every Lie algebra automorphisms of the vector fields $Vec(A^n)$ of affine n-space $A^n$, of the vector fields $Vec^c(A^n)$ with constant divergence, and of the vector fields $Vec^0(A^n)$ with divergence zero is induced by an automorphism of $A^n$. This generalizes results of the second author obtained in dimension 2. The case of $Vec(A^n)$ is due to Vladimir Bavula. As an immediate consequence, we get the following result which due to Viktor Kulikov. If every injective endomorphism of the simple Lie algebra $Vec(A^n)$ is an automorphism, then the Jacobian Conjecture holds in dimension $n$.
The first part of this paper is a refinement of Winkelmann's work on invariant rings and quotients of algebraic groups actions on affine varieties, where we take a more geometric point of view. We show that the (algebraic) quotient $X/\!/\!G$ given by the possibly not finitely generated ring of invariants is "almost" an algebraic variety, and that the quotient morphism $π\colon X \to X/\!/\! G$ has a number of nice properties. One of the main difficulties comes from the fact that the quotient morphism is not necessarily surjective. These general results are then refined for actions of the additive group $\mathbb{G}_a$, where we can say much more. We get a rather explicit description of the so-called plinth variety and of the separating variety, which measures how much orbits are separated by invariants. The most complete results are obtained for representations. We also give a complete and detailed analysis of Roberts' famous example of a an action of $\mathbb{G}_a$ on 7-dimensional affine space with a non-finitely generated ring of invariants.
We show that two varieties X and Y with isomorphic endomorphism semigroups are isomorphic up to field automorphism if one of them is affine and contains a copy of the affine line. A holomorphic version of this result is due to the first author.
We prove a "Generic Equivalence Theorem which says that two affine morphisms $p: S \to Y$ and $q: T \to Y$ of varieties with isomorphic (closed) fibers become isomorphic under a dominant etale base change $ϕ: U \to Y$. A special case is the following result. Call a morphism $ϕ: X \to Y$ a "fibration with fiber $F$" if $ϕ$ is flat and all fibers are (reduced and) isomorphic to $F$. Then an affine fibration with fiber $F$ admits an etale dominant morphism $μ: U \to Y$ such that the pull-back is a trivial fiber bundle: $U\times_Y X \simeq U\times F$. As an application we give short proofs of the following two (known) results: (a) Every affine $\A^1$-fibration over a normal variety is locally trivial in the Zariski-topology; (b) Every affine $\A^2$-fibration over a smooth curve is locally trivial in the Zariski-topology. We also study families of reductive group actions on $\A^2$ parametrized by curves and show that every faithful action of a non-finite reductive group on $Å^3$ is linearizable, i.e. $G$-isomorphic to a representation of $G$.
Let G be a complex reductive group and V a G-module. Let π: V \to V//G be the quotient morphism and set N(V) = π^{-1}(π(0)). We consider the following question. Is the null cone N(V) reduced, i.e., is the ideal of N(V) generated by G-invariant polynomials? We have complete results when G is SL_2, SL_3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.
If V is a representation of a linear algebraic group G, a set S of G-invariant regular functions on V is called separating if the following holds: If two elements v,v' from V can be separated by an invariant function, then there is an f from S such that f(v) is different from f(v'). It is known that there always exist finite separating sets. Moreover, if the group G is finite, then the invariant functions of degree <= |G| form a separating set. We show that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. If G is finite, we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We show that for a subgroup H of G we have b(H) <= b(G) <= [G:H] b(H)$, and that b(G) <= b(G/H) b(H)$ in case H is normal. Moreover, we calculate b(G) for some specific finite groups.