# What is the moduli space of Calabi-Yau manifolds

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?

Thanks.

## 1 Answer

K3 surface is Calabi-Yau manifold. To find a compact space that contains the moduli space of $K3$ surfaces as a dense open set led to compactification of it.

Moduli of $K3$ surfaces can be written as the quotient of the space of Bridgeland stability conditions

Moduli space of polarized $K3$ surfaces of degree 2d, i.e., $K3$ surfaces $X$ which admit an ample divisor $H$ of self-intersection $H^2=2d$. These lie in a 19-dimensional moduli space $F_{2d}$ for each $d>0$. There are several results on compactification of $F_{2d}$, from Baily, Borel and Satake, Laza and Friedman. Also Kollár and N. I. Shepherd-Barron and V. A. Alekseev have described a general method to compactify moduli space $F_{2d}$ using the minimal model program called KSBA compactification.

The moduli space $M_{En}$ of Enriques surfaces(as $K3$ surface) is an open subset of a 10-dimensional orthogonal modular variety, which is rational.

Existence (as finite étale quotient of Hilbert scheme) of moduli space of polarized K3 surfaces and in general polarized CY manifolds studied by E. Viehweg, as a quasi-projective varieties.

Note that the moduli space of K3 surfaces has Kahler metric and canonically we call it Weil-Petersson metric. See this paper

On the moduli space of $log$ CY pairs $(X,D)$, where $K_X+D\cong0$, we don't know yet the existence of moduli space of log-Calabi-Yau pairs. It seems it is difficult question.