Real and complex analytic sets. The relevance of Segre varieties

Abstract

Let X ⊂ C n be a closed real-analytic subset and put A := {z ∈ X |∃ A ⊂ X, germ of a complex-analytic set, z ∈ A, dimz A > 0} This article deals with the question of the structure of A.I nthe main result a natural proof is given for the fact, that A always is closed. As a main tool an interesting relation between complex analytic subsets of X of positive dimension and the Segre varieties of X is proved and exploited.