Discreteness For the Set of Complex Structures On a Real Variety

http://dx.doi.org/10.4153/CMB-2003-033-7
Canad. Math. Bull. 46(2003), 321-322

  Published:2003-09-01
 Printed: Sep 2003

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.