Abstract
We show that the classical Kuga-Satake construction gives rise, away from
characteristic 2, to an open immersion from the moduli of primitively polarized
K3 surfaces (of any fixed degree) to a certain normal integral model for a
Shimura variety of orthogonal type. This allows us to attach to every polarized
K3 surface in odd characteristic an abelian variety such that divisors on the
surface can be identified with certain endomorphisms of the attached abelian
variety. Using a result of Kisin, we can then prove the Tate conjecture for K3
surfaces over finitely generated fields of odd characteristic. We also show
that the moduli stack of primitively polarized K3 surfaces of degree 2d is
quasi-projective and, when d is not divisible by p^2, is geometrically
irreducible in characteristic p. We indicate how the same method applies to
prove the Tate conjecture for co-dimension 2 cycles on cubic fourfolds.
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