Trianalytic subvarieties of the Hilbert scheme of points on a K3 surface

Let X be a hyperkaehler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a K3 surface M, the Hilbert scheme classifying zero-dimensional subschemes of M admits a hyperkaehler structure. We show that for M generic, there are no trianalytic subvarieties of the Hilbert scheme. This implies that a generic deformation of the Hilbert scheme of K3 has no complex subvarieties.