Geodesic
Smooth Maps and Real Algebraic Morphisms
Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 445-451

Voir la notice de l'article provenant de la source Cambridge University Press

Let $X$ be a compact nonsingular real algebraic variety and let $Y$ be either the blowup of ${{\mathbb{P}}^{n}}\left( \mathbb{R} \right)$ along a linear subspace or a nonsingular hypersurface of ${{\mathbb{P}}^{m}}\left( \mathbb{R} \right)\,\times \,{{\mathbb{P}}^{n}}\left( \mathbb{R} \right)$ of bidegree (1, 1). It is proved that a ${{\mathcal{C}}^{\infty }}$ map $f:\,X\,\to \,Y$ can be approximated by regular maps if and only if ${{f}^{*}}\left( {{H}^{1}}\left( Y,\,{\mathbb{Z}}/{2}\; \right) \right)\,\subseteq \,H_{a\lg }^{1}\left( X,\,{\mathbb{Z}}/{2}\; \right)$ , where $H_{a\lg }^{1}\left( X,\,{\mathbb{Z}}/{2}\; \right)$ is the subgroup of ${{H}^{1}}\left( X,\,{\mathbb{Z}}/{2}\; \right)$ generated by the cohomology classes of algebraic hypersurfaces in $X$ . This follows from another result on maps into generalized flag varieties.
DOI : 10.4153/CMB-1999-052-6
Mots-clés : 14P05, 14P25 
Bochnak, J.; Kucharz, W. Smooth Maps and Real Algebraic Morphisms. Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 445-451. doi: 10.4153/CMB-1999-052-6
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