Tame Sets in the Complement of Algebraic Variety |
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Authors: | Dejan Kolari? |
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Institution: | (1) Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia;(2) Institute of Mathematics, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland |
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Abstract: | Let A?? N be an algebraic variety with dim?A≤N?2. Given discrete sequences {a j },{b j }?? N \ A with slow growth ( $\sum_{j}{1\over|a_{j}|^{2}}<\infty,\sum_{j}{1\over |b_{j}|^{2}}<\inftyLet 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 (
?j1/(|aj|2)] < ¥,?j1/(|bj|2)] < ¥\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. |
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Keywords: | |
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