Discreteness for the set of complex structures on a real variety

\Let X, Y be reduced and irreducible compact complex spaces and S the set of all isomorphism classes of reduced and irreducible compact complex spaces W such that X x Y congruent to X x W. Here we prove that S is at most countable. We apply this result to show that for every reduced and irreducible compact complex space X the set S(X) of all complex reduced compact complex spaces W with X x X-sigma congruent to W x W-sigma (where A(sigma) denotes the complex conjugate of any variety A) is at most countable.