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

Abstract

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).

An Oka Principle for a Parametric Infinite Transitivity Property

“…In the particular case when W = C k and X = C n , any map W → Y X,N is null-homotopic, so we recover the result of [16], without any restrictions on the dimension of W . Moreover, Theorem 1.1 reduces the problem of simultaneous standardization of parametrized points in C n by automorphisms in Aut W (C n ) (not the connected component!)…”

“…By the Oka principle (since GL n (C) is Oka-Forstnerič), (16) [Hol(W, GL n (C))] ∼ = [Cont(W, GL n (C))]. As a consequence, the following purely topological characterization of simultaneous standardization can be deduced from our main theorem.…”

“…. , x N are simultaneously standardizable if there exists a "parametrized automorphism" α ∈ Aut W (X), where This notion was introduced by the first author and S. Lodin in [16], where it is shown that for X = C n and W = C k , if k < n − 1, then any collection of parametrized points W → Y X,N is simultaneously standardizable. Our main result is the following.…”

A parametric jet-interpolation theorem for holomorphic automorphisms of C^n

“…We now use [12,Lemma A.6] in order to obtain numbers A, B ∈ N, linear maps {λ j } A j=1 , {µ j } B j=1 ⊂ (C n ) * and vectors {v j } A j=1 , {w j } B j=1 ⊂ C n with λ j (v j ) = 0 and µ j (w j ) = 0 for all j such that the homogeneous polynomial maps of degree r given by z → (λ j (z)) r v j , j = 1, . .…”

“…for uniquely determined holomorphic functions c j , d j : X → C. Recall that the coefficients d j are identically zero if the vector field P r x has vanishing divergence. Thanks to [12,Lemma A.5] we can furthermore ensure that µ j (a i ) = 0 and λ j (a i ) = 0 for all i = 1, . .…”