The Construction of Calabi-Yau Manifolds with Complex Multiplication

In this chapter we explain the basic construction methods of Calabi-yau manifolds with complex multiplication and give a first new example. We call a family of Calabi-Yau n-manifolds, which contains a dense set of fibers X such that the Hodge group of the Hodge structure on H ^k(X,Q) is a torus for all k, a CMCY family of n-manifolds.

In Section 7.1 we explain the technical facts, which we will need for the construction of CMCY families. By using the mirror construction of C. Borcea [9] and C. Voisin [60], we give a method to construct an infinite tower of CMCY families in Section 7.2. In Section 7.3 we discuss the construction method of E. Viehweg and K. Zuo [58]. By using this method given by a tower of cyclic covers, E. Viehweg and K. Zuo [58] have constructed an example of a CMCY family of 3-manifolds. We finish this chapter with the example

$$P^3 \supset V(y_2^4 + y_1^4 + x_1 (x_1 - x_0 )(x_1 - \lambda x_0 )x_0 ) \to \lambda \in M_1$$

of a family of K3 surfaces with a dense set of CM fibers. This example is obtained from the Viehweg-Zuo tower, which starts with the family

$$P^2 \supset V(y_1^4 + x_1 (x_1 - x_0 )(x_1 - \lambda x_0 )x_0 ) \to \lambda \in M_1$$

of curves. This family has a dense set of CM fibers by the previous chapter. By some of its involutions the family of K3 surfaces above is suitable for the construction of a Borcea-Voisin tower.