Transcendental submanifolds of projective space

Abstract

Given integers $m$ and $c$ satisfying $m-2\ge c\ge 2$, we explicitly construct a nonsingular $m$-dimensional algebraic subset of $P^{m+c}(R)$ that is not isotopic to the set of real points of any nonsingular complex algebraic subset of $P^{m+c}(C)$ defined over $R$. The first examples of this type were obtained by Akbulut and King in a more complicated and nonconstructive way, and only for certain large integers $m$ and $c$.