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The Construction of Calabi-Yau Manifolds with Complex Multiplication

  • Jan Christian RohdeEmail author
Chapter
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Part of the Lecture Notes in Mathematics book series (LNM, volume 1975)

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 Hk(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.

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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Institut fuer Algebraische Geometrie Leibniz Universität Hannover Welfengarten 1, GRK 1463HannoverGermany

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