TRIANALYTIC SUBVARIETIES OF THE HILBERT SCHEME OF POINTS ON A K3 SURFACE

Let X be a hyperkahler manifold. Trianalytic subvarieties of X are sub varieties which are complex analytic with respect; to all complex stru ctures induced by the hyperkahler structure. Given a K3 surface ill, t he Hilbert scheme classifying zero-dimensional subschemes of M admits a hyperkahler structure. We show that fur nl generic, there are no tri analytic subvarieties of the Hilbert scheme. This implies that a gener ic deformation of the Hilbert scheme of K3 has no proper complex subva rieties.