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Transcendental submanifolds of projective space. (English) Zbl 1209.57023

Summary: Given integers \(m\) and \(c\) satisfying \(m - 2 \geq c \geq 2\), we explicitly construct a nonsingular \(m\)-dimensional algebraic subset of \(\mathbb P^{m + c}(\mathbb R)\) that is not isotopic to the set of real points of any nonsingular complex algebraic subset of \(\mathbb P^{m + c}(\mathbb C)\) defined over \(\mathbb R\). 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 \(c\).

MSC:

57R55 Differentiable structures in differential topology
14P25 Topology of real algebraic varieties
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