Tame sets in the complement of algebraic variety
Tame sets in \(\mathbb C^n\) are sequences which can be mapped into a complex line by some holomorphic automorphism of \(\mathbb C^n\). Given an algebraic variety \(A\subset \mathbb C^n\) with codimension\(>2\), a sequence in \(\mathbb C^n\) is called an \(A\)-tame set if it can be mapped into a complex line by a holomorphic automorphism of \(\mathbb C^n\) fixing all the points of \(A\). According to the main result, Theorem 2.3, any sequence \(\{ a_j\}\) of distinct points in \(\mathbb C^n\) with the slow growth property \(\sum_j | a_j|^{-2}<\infty\) is an \(A\)-tame set, moreover given another sequence \(\{ b_j\}\) with slow growth, one can construct a holomorphic automorphism \(F:\mathbb C^n\to\mathbb C^n\) fixing the points of \(A\) such that \(F(a_j)=b_j\) \((j=1,2,\dots)\). The construction allows even more constraints: given any polynomially convex compact set \(K\subset\mathbb C^n\) along with a holomorphic automorphism \(G\mathbb C^n\to\mathbb C^n\) fixing \(A\) such that \(K\cap \{ a_j\},G(K)\cap \{ b_j\} =\emptyset\), the mapping \(F\) above can be chosen to approximate \(G\) on \(K\). On the other hand, if \(A\) is an unbounded proper analytic variety, then there exists a tame set which cannot be mapped into an arithmetic sequence by any holomorphic automorphism of \(\mathbb C^n\) fixing \(A\). As an application the author constructs a proper injective holomorphic embedding of an \(n\)-dimensional Stein manifold \(X\) into \(\mathbb C^N \setminus (A\cup B)\) where where \(N:=n+[n/2]+1\), \(A\) is an analytic variety in \(\mathbb C^N\) of dimension \(\leq N-([n/2]+1)\) and \(B\) is the unit ball of \(\mathbb C^N\). Finally this latter result is refined with interpolation on a given discrete set.
scientific article; zbMATH DE number 5624082
scientific article