  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.
