Regular separation with parameter of complex analytic sets

Abstract

The aim of this paper is to prove that a pair of analytic sets X, Y ⊂ C_z^m × C_wⁿ is locally regularly separated with a uniform exponent α in the fibres taken over a proper projection π(z,w) = z of X ∩ Y (under the assumption that X ∩ Y has pure dimension): for all z ∈ π (X ∩ Y) ∩ U, dist(w,Y^z) ≥ const.dist(w,(X ∩ Y)^z)^α when w ∈ X^z ∩ V, where U × V is a neighbourhood of a point a ∈ X ∩ Y such that π(a) is regular in π(X ∩ Y). As an application of this we obtain a parameter version of the Łojasiewicz inequality for c-holomorphic mappings. Both results are a complex counterpart of the main result of [ŁW] from the subanalytic case, extended in this paper by a bound on the uniform exponent.