Transcendental submanifolds of projective space

Abstract

Given integers m and c satisfying m − 2 ≥ c ≥ 2, we explicitly construct a nonsingular m-dimensional algebraic subset of ℙm + c(ℝ) that is not isotopic to the set of real points of any nonsingular complex algebraic subset of ℙm + 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.