 
Summary: arXiv:math.AG/0404463v21Dec2004
TRANSCENDENTAL SUBMANIFOLDS OF RPn
SELMAN AKBULUT AND HENRY KING
Abstract. In this paper we give examples of closed smooth submanifolds of RPn
which are isotopic to nonsingular projective subvarieties of RPn
but they can not
be isotopic to the real parts of nonsingular complex projective subvarieties of CPn
.
0. Introduction
Let j : Rn
RPn
be the canonical imbedding as a chart. Real algebraic sets in Rn
are not in general real algebraic sets in RPn
. The Zariski closure of the image of an
algebraic set (under j) usually has extra components at infinity. An algebraic subset
of Rn
which remains an algebraic set in RPn
is called a projectively closed algebraic
set ([AK1]). Not every algebraic set is projectively closed. In general, isotoping a
submanifold of the projective space RPn
