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# 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 $X\times Y \cong X\times W$. 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 $X\times X^\sigma \cong W\times W^\sigma$ (where $A^\sigma$ denotes the complex conjugate of any variety $A$) is at most countable.
 MSC Classifications: 32J18 - Compact $n$-folds 14J99 - None of the above, but in this section 14P99 - None of the above, but in this section