Rational surfaces and regular maps into the 2-dimensional sphere

Abstract

Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a \({\cal C}^\infty\) mapping, \(f:X\rightarrow Y\), can be approximated by regular mappings in the space of \({\cal C}^\infty\) mappings, \({\cal C}^\infty(X,Y)\), equipped with the \({\cal C}^\infty\) topology. In this paper, we obtain some results concerning this problem when the target space is the 2-dimensional standard sphere and X has a complexification \(X_{\mathbb C}\) that is a rational (complex) surface. To get the results we study the subgroup \(H^2_{{\mathbb C}-\text{alg}}(X,{\mathbb Z})\) of the second cohomology group of X with integer coefficients that consists of the cohomology classes that are pullbacks, via the inclusion mapping \(X\rightarrow X_{\mathbb C}\), of the cohomology classes in \(H^2(X_{\mathbb C},{\mathbb Z})\) represented by complex algebraic hypersurfaces.