ON THE REAL TRACES OF ANALYTIC VARIETIES

Let V be a complex-analytic variety of complex dimension n which is the complexification of its real trace V⁰, so that V⁰ is a real-analytic variety of real dimension n, and let X be a complex-analytic subvariety of complex dimension r in V such that X⁰ = X ∩ V⁰ has the real dimension r (and hence ). If we denote by ϕ(X) and ϕ(V⁰) the homology classes mod 2 determined by X and V⁰ respectively as topological cycles mod 2 in V, then their intersection class ϕ (X) · ϕ (V⁰), with due regard for their supports, can be considered as an element in H_2r-n(V⁰, Z₂). On the other hand, X⁰ is also a topological cycle mod 2 in V⁰ and hence determines an element ψ(X⁰) in H_r(V°, Z₂). It is natural to ask about the relation between these two elements ϕ(X) · ϕ(V°) and ψ(X⁰) in H_*(V⁰, Z₂), since roughly speaking they represent respectively the algebraic intersection and the geometric intersection of X and V⁰. We shall give here an answer to this question for the case where V and X are algebraic…