Approximation of analytic sets along Nash subvarieties

Abstract

Let X be an analytic subset of pure dimension n of an open set U ⊂ C m and let E be a Nash subset of U such that E ⊂ X.Then for every a ∈ E there is an open neighborhood V of a in U and a sequence { X v } of complex Nash subsets of V of pure dimension n converging to X ∩ V in the sense of holomorphic chains such that the following hold for every v ∈ N: E ∩ V ⊂ X v and the multiplicity of X v at x equals the multiplicity of X at x for every x in a dense open subset of E ⊂ V.