# The Construction of Calabi-Yau Manifolds with Complex Multiplication

Chapter

First Online:

- 737 Downloads

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 of a family of of curves. This family has a dense set 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$$P^3 \supset V(y_2^4 + y_1^4 + x_1 (x_1 - x_0 )(x_1 - \lambda x_0 )x_0 ) \to \lambda \in M_1$$

*K*3 surfaces with a dense set of*CM*fibers. This example is obtained from the Viehweg-Zuo tower, which starts with the family$$P^2 \supset V(y_1^4 + x_1 (x_1 - x_0 )(x_1 - \lambda x_0 )x_0 ) \to \lambda \in M_1$$

*CM*fibers by the previous chapter. By some of its involutions the family of*K*3 surfaces above is suitable for the construction of a Borcea-Voisin tower.## Keywords

Elliptic Curve Complex Multiplication Elliptic Curf Hodge Structure Cyclic Cover
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Copyright information

© Springer-Verlag Berlin Heidelberg 2009