Smooth maps and real algebraic morphisms

Let X be a compact nonsingular real algebraic variety and let Y be either the blowup of P-n(R) along a linear subspace or a nonsingular hypersurface of P-m(R) x P-n(R) of bidegree (1, 1). It is proved that a ---infinity map f: X --> Y can be approximated by regular maps if and only if f* (H-1(Y, Z/2)) subset of or equal to H-alg(1)(X, Z/2)' where H-alg(1) (X, Z/2) is the subgroup of H-1(X, Z/2) generated by the cohomology classes of algebraic hypersurfaces in X. This follows from another result on maps into generalized flag varieties.