Tame Sets in the Complement of Algebraic Variety

Abstract

Let A⊂ℂ^N be an algebraic variety with dim A≤N−2. Given discrete sequences {a _j },{b _j }⊂ℂ^N \ A with slow growth ( $\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\infty$ ) we construct a holomorphic automorphism F with F(z)=z for all z∈A and F(a _j )=b _j for all j∈ℕ. Additional approximation of a given automorphism on a compact polynomially convex set, fixing A, is also possible. Given unbounded analytic variety A there is a tame set E such that F(E)≠{(j,0^N−1):j∈ℕ} for all automorphisms F with F| _A =id. As an application we obtain an embedding of a Stein manifold into the complement of an algebraic variety in ℂ^N with interpolation on a given discrete set.