Smooth Maps and Real Algebraic Morphisms

http://dx.doi.org/10.4153/CMB-1999-052-6
Canad. Math. Bull. 42(1999), 445-451

  Published:1999-12-01
 Printed: Dec 1999

Let $X$ be a compact nonsingular real algebraic variety and let $Y$ be either the blowup of $\mathbb{P}^n(\mathbb{R})$ along a linear subspace or a nonsingular hypersurface of $\mathbb{P}^m(\mathbb{R}) \times \mathbb{P}^n(\mathbb{R})$ of bidegree $(1,1)$. It is proved that a $\mathcal{C}^\infty$ map $f \colon X \rightarrow Y$ can be approximated by regular maps if and only if $f^* \bigl( H^1(Y, \mathbb{Z}/2) \bigr) \subseteq H^1_{\alg} (X,\mathbb{Z}/2)$, where $H^1_{\alg} (X,\mathbb{Z}/2)$ is the subgroup of $H^1 (X, \mathbb{Z}/2)$ generated by the cohomology classes of algebraic hypersurfaces in $X$. This follows from another result on maps into generalized flag varieties.