Rational Hodge isometries of hyper-Kähler varieties of type are algebraic

Let X and Y be compact hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. A class in H^p, p (X Y, {Q}) is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let f: H² (X, {Q}) H² (Y, {Q}) be a rational Hodge isometry with respect to the Beauville–Bogomolov–Fujiki pairings. We prove that f is induced by an analytic correspondence. We furthermore lift f to an analytic correspondence f: H^* (X, {Q}) 2n H^* (Y, {Q}) 2n, which is a Hodge isometry with respect to the Mukai pairings and which preserves the gradings up to sign. When X and Y are projective, the correspondences f and f are algebraic.