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 Hp,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 f̃: 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 f̃ are algebraic.