Algebraic approximations in analytic geometry (Q1899737)

Let \(V\) and \(W\) be affine algebraic varieties and \(K\) be a compact holomorphically convex subset of \(V\). It is proved that a holomorphic mapping \(f : K \to W\) can be uniformly approximated on \(K\) by Nash algebraic mappings \(g : K \to W\) (i.e. mappings whose components \(g_j\) satisfy polynomial equations \(P_j (z, g_j (z)) = 0)\). The result is used to prove that a reduced Stein space with isolated singularities can be exhausted by open sets that are biholomorphic to open sets in affine algebraic varieties. As a consequence it is shown that if a strongly pseudoconvex, compact CR manifold bounds a strongly pseudoconvex, compact complex manifold, then it also bounds a strongly pseudoconcave, compact complex manifold.