Approximation of analytic sets with proper projection by Nash sets

Approximation of Analytic Sets with Proper Projection by Algebraic Sets

“…Proving Theorem 3.1, we considerably strengthen the results of [4,5], where it is shown that purely k-dimensional analytic subsets of U × C n with proper projection onto a Runge domain U ⊂ C k can be approximated by complex Nash sets. The latter fact is the starting point for our considerations.…”

“…Proof of Theorem 3.1 Let us first recall that for analytic covers, there exist Nash approximations: Papers [4,5] contain detailed proofs of Theorem 3.5. Here let us just mention that this theorem is related to the problem of approximation of holomorphic maps between complex (algebraic) spaces, for which the reader is referred to [1,9,13,14,18,19,[25][26][27].…”

Approximation of analytic sets by Nash tangents of higher order

“…The set E determines the relations which, among other things, guarantee the appropriate dimension of the approximating sets: the set X can be represented by the mapping ψ : B m (1) → E (cp. the proof of Theorem 1.1 in [4]. For the way X is recovered from ψ see the second paragraph of the proof of Theorem 4 in the sequel).…”

“…The idea of algebraic approximation of analytic objects in complex geometry has been developed by several mathematicians (see [3][4][5]9,11,[17][18][19]). In this paper we investigate the approximation of complex analytic sets by complex Nash sets which is expressed in terms of the convergence of holomorphic chains.…”

“…In this paper we investigate the approximation of complex analytic sets by complex Nash sets which is expressed in terms of the convergence of holomorphic chains. The existence of such approximation for sets with proper projection was established in [3,4]. In particular in [4] it was proved that for every point a ∈ X, where X is any locally analytic subset of C m , there exist an open neighborhood V of a in C m and a sequence of complex Nash subsets of V converging to X ∩ V. (For the notion of chains and their convergence see Sect.…”