On the space of real algebraic morphisms

Ghiloni, Riccardo

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 4, pp. 307-317

Résumé

In this Note, we announce several results concerning basic properties of the spaces of morphisms between real algebraic varieties. Our results show a surprising intrinsic rigidity of Real Algebraic Geometry and illustrate the great distance which, in some sense, exists between this geometry and Real Nash one. Let us give an example of this rigidity. An affine real algebraic variety $X$ is rigid if, for each affine irreducible real algebraic variety $Z$, the set of all nonconstant regular morphisms from $Z$ to $X$ is finite. We are able to prove that, given a compact smooth manifold $M$ of positive dimension, there exists an uncountable family $\{ M_{i}\}_{i \in I}$ of rigid affine nonsingular real algebraic varieties diffeomorphic to $M$ such that, for each $i \neq j$ in $I$, $M_{i}$ is not biregularly isomorphic to $M_{j}$.