Global almost analytic algebraicity of analytic sets

Abstract

1. The main result. In this paper real algebraic varieties and morphisms between them are understood in the sense of Serre [15] (Serre considers algebraic varieties over an algebraically closed field but his basic definitions make sense over any field). The reader may consult a detailed exposition [7] for properties of real algebraic varieties, especially in connection with real algebraic blow-ups (cf. also [1]). All subvarieties will be assumed closed but not necessarily irreducible. Let Y be an aίfine real algebraic variety and let Z be a subvariety of Y. Then the algebraic blow-up π:B —• Y of Y along Z has the following properties: B is an affine real algebraic variety, π is a real algebraic morphism whose restriction to B\π~λ(Z) is an algebraic isomorphism onto Y\Z, and π is a proper map if B and Y are equipped with the Euclidean topology. Moreover, B is nonsingular and π is surjective if Y and Z are nonsingular varieties. Let X and X be affine nonsingular real algebraic varieties and let D be a subvariety of X. An algebraic morphism π: X —• X is called a k-fold algebraic multiblowup of X along D if π is the composition π = m o o nk , where v v π* Y %k~x π2 Y πi Y V