Complex Multiplication, Rationality and Mirror Symmetry for Abelian Varieties and K3 Surfaces

Abstract

This thesis consists of three parts. In the first part we study abelian varieties and K3 surfaces of CM-type (i.e. their Hodge group is commutative), aiming at a characterization of complex multiplication via the existence of special Kähler metrics. We find out that an abelian variety is of CM-type if and only if it admits a rational Kähler metric. For K3 surfaces of CM-type, some arithmetic properties can also be formulated.<br /> In the second part we apply the characterizations we found above in order to give sufficient conditions under which a mirror of an abelian variety or of a K3 surface of CM-type is of CM-type as well.<br /> In the third part we construct a lattice superconformal OPE-algebra and define the notion of rationality for it. Then we study the rationality of this algebra once associated to an abelian variety of CM-type and also in the case where a mirror is of CM-type.