Rational Hodge isometries of hyper-Kahler varieties of K3[n] type are algebraic

Let X and Y be compact hyper-Kahler manifolds deformation equivalent to the Hilbert scheme of length n subschemes of a K3 surface. A class in H-p,H-p (X x Y, Q) is an analytic correspondence, if it belongs to the subring generated by Chern classes of coherent analytic sheaves. Let f:H-2(X,Q) -> H-2(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) over tilde are algebraic.