Nonlinearizable holomorphic group actions

Derksen, Harm; Kutzschebauch, Frank

doi:10.1007/s002080050175

Cited by 34 publications

(35 citation statements)

References 10 publications

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“…It was a long standing problem whether all holomorphic actions of such groups on affine space are linear after a change of variables (see for example the overview article [28]). The first counterexamples to that (Holomorphic Linearization) problem were constructed by Derksen and the first author in [8]. In the present paper we show that the method from there is holomorphic in a parameter and therefore applied to our parametrized situation leads to Theorem 1.3.…”

Section: Introduction and Statement Of The Main Resultssupporting

confidence: 50%

“…The later biholomorphism comes from the fact that there is an automorphism of C n+l+1 leaving the divisor {v = 0} invariant and straightening the center ϕ(C l ) × 0 v inside the divisor (see Lemma 2.5. in [8]). …”

Section: Families Of Holomorphic C * -Actions On Affine Spacementioning

confidence: 98%

“…In this section we employ the method from [8] and [9] to construct (non-linearizable) C * -actions on affine spaces out of embeddings C l ֒→ C n . We will not give all proofs in detail.…”

Section: Families Of Holomorphic C * -Actions On Affine Spacementioning

confidence: 99%

“…which in [8] is called Rees space. This notion was introduced there by the authors since they were not aware of the fact that this is a wellknown construction, called affine modification, going back to Oscar Zariski.…”

Section: Families Of Holomorphic C * -Actions On Affine Spacementioning

confidence: 99%

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Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

Kutzschebauch¹,

Lodin²

2013

Duke Math. J.

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Abstract. We construct holomorphic families of proper holomorphic embeddings of C k into C n (0 < k < n− 1), so that for any two different parameters in the family no holomorphic automorphism of C n can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of C n we derive the existence of families of holomorphic C * -actions on C n (n ≥ 5) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in particular asked whether there would be more than one conjugacy class of C * actions on C n (with prescribed linear part at a fixed point).

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Section: Introduction and Statement Of The Main Resultssupporting

confidence: 50%

Section: Families Of Holomorphic C * -Actions On Affine Spacementioning

confidence: 98%

Section: Families Of Holomorphic C * -Actions On Affine Spacementioning

confidence: 99%

Section: Families Of Holomorphic C * -Actions On Affine Spacementioning

confidence: 99%

See 2 more Smart Citations

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

Kutzschebauch¹,

Lodin²

2013

Duke Math. J.

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“…Thus the algebraic counterexamples to linearisation are not counterexamples in the holomorphic category. But Derksen and Kutzschebauch [DK98] showed that for G nontrivial, there is an N G ∈ N such that there are nonlinearisable holomorphic actions of G on C n , for every n ≥ N G .…”

Section: Introductionmentioning

confidence: 99%

Sufficient Conditions for Holomorphic Linearisation

Kutzschebauch

Lárusson

Schwarz

2016

Transformation Groups

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Abstract. Let G be a reductive complex Lie group acting holomorphically on X = C n . The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on C n such that the G-action becomes linear. Equivalently, is there a Gequivariant biholomorphism Φ : X → 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 : Q X → 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 C n , only that X is a Stein manifold.

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Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

Kutzschebauch¹

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka-Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930's, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview article we present 3 classes of properties: 1. density property, 2. flexibility 3. Oka-Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

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Nonlinearizable holomorphic group actions

Cited by 34 publications

References 10 publications

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

Sufficient Conditions for Holomorphic Linearisation

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

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