This is supposed to be a generic question for Calabi-Yau manifolds, but for definiteness let me exemplify it with the $K3$ manifold. What is the moduli space of $K3$ manifolds? I am also asking what really means "moduli space of $K3$ smanifold". What does it represent? Are there several nonequivalent notions of moduli space of $K3$ manifold? Are the moduli space of $K3$ manifolds and the moduli space of complex $K3$ surfaces the same?
In String Theory is very much used that the moduli space of a Calabi-Yau three-fold is locally a product of two Special Kahler manifolds, which matches the Supergravity prediction through the geometry of the corresponding non-linear sigma model (as it should happen). However, I have failed to find a mathematical reference who speaks of the moduli space of Calabi-Yau manifolds in the same terms. How is this Special Kahler geometry of the moduli space of Calabi-Yau three-folds proven?