Abstract

Given a polarized complex K3 surface, one can attach to it a complex abelian variety,
called Kuga-Satake variety. The Kuga-Satake variety is determined by the singular
cohomology of the K3 surface; on the other hand, this singular cohomology can be
recovered by means of the weight 1 Hodge structure associated to the Kuga-Satake
variety. Despite the transcendental origin of this construction, Kuga-Satake varieties
have interesting arithmetic properties. Kuga-Satake varieties of K3 surfaces defined
over number fields descend to finite extension of the field of definition. This property
suggests that the Kuga-Satake construction can be interpreted as a map between
moduli spaces. More precisely, one can define a morphism, called Kuga-Satake map,
between the moduli space of K3 surfaces and the moduli space of abelian varieties
with polarization and level structure. This morphism, defined over a number field,
is obtained by regarding the classical construction as a map between an orthogonal
Shimura variety, closely related to the moduli space of K3 surfaces, and the Siegel
modular variety. The most remarkable fact is that the Kuga-Satake map extends
to positive characteristic for almost all primes, associating to K3 surfaces abelian
varieties over finite fields. This can be proven applying a result by Faltings on the
extension of abelian schemes and the good reduction property of Kuga-Satake varieties.