Complex multiplication, rationality and mirror symmetry for abelian varieties

Abstract

We show that complex multiplication on abelian varieties is equivalent to the existence of a constant rational K\"ahler metric. We give a sufficient condition for a mirror of an abelian variety of CM-type to be of CM-type as well. We also study the relationship between complex multiplication and rationality of a toroidal lattice vertex algebra.Comment: 22 pages, introduction and exposition of results rewritten, references adde