Smooth Maps and Real Algebraic Morphisms

Abstract. Let X be a compact nonsingular real algebraic variety and let Y be either the blowup of Pn (R)along a linear subspace or a nonsingular hypersurface of Pm (R) × Pn (R) ofbidegree(1,1). It is proved that a C ∞ map f: X → Y can be approximated by regular maps if and only if f ∗ ( H1 (Y, Z/2) ) ⊆ H1 alg (X, Z/2), where H1 alg (X, Z/2) is the subgroup of H1 (X, Z/2) generated by the cohomology classes of algebraic hypersurfaces in X. This follows from another result on maps into generalized flag varieties. Throughout this note the term real algebraic variety designates a locally ringed space isomorphic to a Zariski locally closed subset of P n (R), for some n, endowed with the Zariski topology and the sheaf of R-valued regular functions. It is well known that every real algebraic variety is isomorphic to a Zariski closed subvariety of R n for some n [2, Proposition 3.2.10, Theorem 3.4.4] (note that real algebraic varieties defined above are called affine real algebraic varieties in [2]). Morphisms between real algebraic varieties will be called regular maps. Every real algebraic variety carries also the Euclidean topology, that is, the topology induced by the usual metric topology on R. Unless explicitly stated otherwise, all topological notions related to real algebraic varieties will refer to the Euclidean topology. Given two nonsingular real algebraic varieties X and Y, we regard the set R(X,Y)of all regular maps from X into Y as a subset of the space C ∞ (X,Y)ofallC ∞ maps from