  Kuga and Satake associate with every polarized complex K3 surface (X,L) a
complex abelian variety called the Kuga-Satake abelian variety of (X,L). We use
this construction to define morphisms between moduli spaces of polarized K3
surface with certain level structures and moduli spaces of polarized abelian
varieties with level structure over C. In this note we study these morphisms.
We prove first that they are defined over finite extensions of Q. Then we show
that they extend in positive characteristic. In this way we give an indirect
construction of Kuga-Satake abelian varieties over an arbitrary base. We also
give some applications of this construction to canonical lifts of ordinary K3
surfaces.
