Given a K3 surface, M. Kuga and I. Satake [13] associate to it an abelian variety. P. Deligne [5] has constructed an absolute Hodge cycle on the product of the abelian variety and the K3 surface. When the K3 surface is the desingularisation of a double cover of the plane branched along six lines in general position, the abelian variety is isogenous to the product of a number of copies of a four dimensional abelian variety. We construct the Kuga-Satake-Deligne correspondence between this four dimensional abelian variety and the K3 surface in this case.
More precisely, we prove
Theorem 2.1. Let Y 1 be the desingularisation of the double cover of the plane branched along six lines in general position. Then there is an abelian variety P and a codimension two algebraic cycle on P × Y 1 such that the homomorphism
is an inclusion when restricted to the lattice of transcendental cycles of Y 1, and hence this cycle represents the Kuga-Satake-Deligne correspondence.
This result settles affirmatively a special case of the Hodge conjecture.