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Frank Kutzschebauch

Publications and source records attributed to Frank Kutzschebauch.

At least 37 records · Page 2Linked to original sources
arXiv2021

Linearization of holomorphic families of algebraic automorphisms of the affine plane

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.

math.AG↗
arXiv2021

A characterization of linearizability for holomorphic $\mathbb{C}^*$-actions

Let $G$ be a reductive complex Lie group acting holomorphically on $X=\mathbb{C}^n$. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on $\mathbb{C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $Φ\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $X /\!/G$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $Φ$ as above. Then $Φ$ induces a biholomorphism $ϕ\colon X/\!/G\to V/\!/G$ which is stratified, i.e., the stratum of $ X/\!/G$ with a given label is sent isomorphically to the stratum of $V/\!/G$ with the same label. The counterexamples to the Linearization Problem construct an action of $G$ such that $X/\!/G$ is not stratified biholomorphic to any $V/\!/G$. Our main theorem shows that, for a reductive group $G$ with $G^0=\mathbb{C}^*$, the existence of a stratified biholomorphism of $X/\!/G$ to some $V/\!/G$ is not only necessary but also sufficient for linearization. In fact, we do not have to assume that $X$ is biholomorphic to $\mathbb{C}^n$, only that $X$ is a Stein manifold.

math.CV↗
arXiv2019

Complete algebraic vector fields on affine surfaces

Let $\AAutH (X)$ be the subgroup of the group $\AutH (X)$ of holomorphic automorphisms of a normal affine algebraic surface $X$ generated by elements of flows associated with complete algebraic vector fields. Our main result is a classification of all normal affine algebraic surfaces $X$ quasi-homogeneous under $\AAutH (X)$ in terms of the dual graphs of the boundaries $\bX \setminus X$ of their SNC-completions $\bX$.

math.CV↗
arXiv2019

Factorization of symplectic matrices into elementary factors

We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic matrices with entries in a Banach algebra or in the ring of complex valued continuous functions on a finite dimensional normal topological space.

math.SG↗
arXiv2019

Exponential factorizations of holomorphic maps

We show that any element of the special linear group $SL_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $GL_2(R)$.

math.CV↗
arXiv2019

Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem

We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Andersén and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered by many authors, has had a strong influence on those studies. It asks whether a compact subgroup in the holomorphic automorphism group of $\C^n$ is necessarily conjugate to a group of linear automorphisms. Despite many positive results, the answer in this generality is negative as shown by Derksen and the author. We describe various developments around that problem.

math.CV↗
arXiv2019

Factorization by elementary matrices, null-homotopy and products of exponentials for invertible matrices over rings

Let $R$ be a commutative unital ring. A well-known factorization problem is whether any matrix in $\mathrm{SL}_n(R)$ is a product of elementary matrices with entries in $R$. To solve the problem, we use two approaches based on the notion of the Bass stable rank and on construction of a null-homotopy. Special attention is given to the case, where $R$ is a ring or Banach algebra of holomorphic functions. Also, we consider a related problem on representation of a matrix in $\mathrm{GL}_n(R)$ as a product of exponentials.

math.AC↗
arXiv2017

Homotopy principles for equivariant isomorphisms

Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. When is there an equivariant biholomorphism of $X$ and $Y$? A necessary condition is that the categorical quotients $Q_X$ and $Q_Y$ are biholomorphic and that the biholomorphism $ϕ$ sends the Luna strata of $Q_X$ isomorphically onto the corresponding Luna strata of $Q_Y$. Fix $ϕ$. We demonstrate two homotopy principles in this situation. The first result says that if there is a $G$-diffeomorphism $Φ\colon X\to Y$, inducing $ϕ$, which is $G$-biholomorphic on the reduced fibres of the quotient mappings, then $Φ$ is homotopic, through $G$-diffeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. The second result roughly says that if we have a $G$-homeomorphism $Φ\colon X\to Y$ which induces a continuous family of $G$-equivariant biholomorphisms of the fibres $p_X^{-1}(q)$ and $p_Y^{-1}(ϕ(q))$ for $q\in Q_X$ and if $X$ satisfies an auxiliary property (which holds for most $X$), then $Φ$ is homotopic, through $G$-homeomorphisms satisfying the same conditions, to a $G$-equivariant biholomorphism from $X$ to $Y$. Our results improve upon earlier work of the authors and use new ideas and techniques.

math.CV↗
arXiv2017

On subelliptic manifolds

A smooth complex quasi-affine algebraic variety $Y$ is flexible if its special group $\SAut (Y)$ of automorphisms (generated by the elements of one-dimensional unipotent subgroups of $\Aut (Y)$) acts transitively on $Y$. An irreducible algebraic manifold $X$ is locally stably flexible if it is the union $\bigcup X_i$ of a finite number of Zariski open sets, each $X_i$ being quasi-affine, so that there is a positive integer $N$ for which $X_i\times \mathbb{C}^N$ is flexible for every $i$. The main result of this paper is that the blowup of a locally stably flexible manifold at a smooth algebraic submanifold (not necessarily equi-dimensional or connected) is subelliptic, and hence Oka. This result is proven as a corollary of some general results concerning the so-called $k$-flexible manifolds.

math.AG↗
arXiv2016

An equivariant parametric Oka principle for bundles of homogeneous spaces

We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$ over $X$ into the space of $K$-equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describe several diverse special cases. We use the result to strengthen Heinzner and Kutzschebauch's classification of equivariant principal bundles, and to strengthen an Oka principle for equivariant isomorphisms proved by us in a previous paper.

math.CV↗
arXiv2016

The density property for Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$

Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$ can be described by the equations $yu = x P(x)$, $xv = u Q(u)$ and $yv = P(x) Q(u)$ in $\mathbb{C}^4_{x,y,u,v}$ where $P$ and $Q$ are non-constant polynomials. We establish the algebraic density property for smooth Gizatullin surfaces of this type. Moreover we also prove the density property for smooth surfaces given by these equations when $P$ and $Q$ are holomorphic functions.

math.CV↗
arXiv2016

The fibred density property and the automorphism group of the spectral ball

We generalize the notion of the density property for complex manifolds to holomorphic fibrations, and introduce the notion of the fibred density property. We prove that the natural fibration of the spectral ball over the symmetrized polydisc enjoys the fibred density property and describe the automorphism group of the spectral ball.

math.CV↗
arXiv2016

Sufficient Conditions for Holomorphic Linearisation

Let $G$ be a reductive complex Lie group acting holomorphically on $X={\mathbb C}^n$. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ${\mathbb C}^n$ such that the $G$-action becomes linear. Equivalently, is there a $G$-equivariant biholomorphism $Φ\colon X\to V$ where $V$ is a $G$-module? There is an intrinsic stratification of the categorical quotient $Q_X$, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of $G$. Suppose that there is a $Φ$ as above. Then $Φ$ induces a biholomorphism $ϕ\colon Q_X\to Q_V$ which is stratified, i.e., the stratum of $Q_X$ with a given label is sent isomorphically to the stratum of $Q_V$ with the same label. The counterexamples to the Linearisation Problem construct an action of $G$ such that $Q_X$ is not stratified biholomorphic to any $Q_V$. Our main theorem shows that, for most $X$, a stratified biholomorphism of $Q_X$ to some $Q_V$ is sufficient for linearisation. In fact, we do not have to assume that $X$ is biholomorphic to ${\mathbb C}^n$, only that $X$ is a Stein manifold.

math.CV↗
arXiv2015

An Oka Principle for a Parametric Infinite Transitivity Property

It is an elementary fact that the action by holomorphic automorphisms on C^n is infinitely transitive, i.e., m-transitive for any m in N. The same holds on any Stein manifold with the holomorphic density property X. We study a parametrized case: we consider m points on X parametrized by a Stein manifold W, and seek a family of automorphisms of X, parametrized by W, putting them into a standard form which does not depend on the parameter. This general transitivity is shown to enjoy an Oka principle, to the effect that the obstruction to a holomorphic solution is of a purely topological nature. In the presence of a volume form and of a corresponding density property, similar results for volume-preserving automorphisms are obtained.

math.CV↗
arXiv2015

Algebraic (Volume) Density Property for Affine Homogeneous Spaces

Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\C$. (1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant volume form $ω$. We prove that the space of all divergence-free (with respect to $ω$) algebraic vector fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on $X$ (including the cases when $X$ is a line or a torus). The proof of these results requires new criteria for algebraic (volume) density property based on so called module generating pairs.

math.CV↗
arXiv2014

Lie algebra generated by locally nilpotent derivations on Danielewski surfaces

We give a full description of the Lie algebra generated by locally nilpotent derivations (short LNDs) on smooth Danielewski surfaces $D_p$ given by $xy=p(z)$. In case $\mathrm{deg}(p)\geq 3$ it turns out to be not the whole Lie algebra $\mathrm{VF}_{alg}^ω(D_p)$ of volume preserving algebraic vector fields, thus answering a question posed by Lind and the first author. Also we show algebraic volume density property (short AVDP) for a certain homology plane, a homogeneous space of the form $SL_2 (\mathbb{C}) /N$, where $N$ is the normalizer of the maximal torus and another related example. At the end of the paper we show by example that for the group of holomorphic automorphisms of a Stein manifold (endowed with c.-o. topology) the connected component and the path-connected component of the identity may not coincide.

math.CV↗