Sixfolds of generalized Kummer type and K3 surfaces

I will present a construction which associates to any sixfold K of generalized Kummer type a hyper-Kähler manifold Y deformation equivalent to a Hilbert scheme of lenght 3 subscheme on a K3 surface, relating the most well studied deformation types of hyper-Kähler manifolds in dimension 6. Our construction is reminiscent of the classical construction of Kummer K3 surfaces, in the sense that Y is obtained as resolution of the quotient of K by a group of symplectic automorphisms. As a consequence we are able to show that any projective sixfold K as above determines a well-defined K3 surface. We use this construction to prove that the Kuga-Satake correspondence is algebraic for infinitely many new families of K3 surfaces of general Picard rank 16.