Let S be a smooth projective surface over the complex numbers; let S-(r) be its r-fold symmetric product and S-[r]] the Hilbert scheme of O-dimensional subschemes of length r.
In case K-S is trivial, the deformation theory of S[PI has been studied by Beauville and Fujiki in order to construct examples of higher-dimensional symplectic manifolds. In that case S-[r] has deformations which are not Hilbert schemes of points on a surface.
We prove that under suitable hypotheses (e.g, if S is of general type) this cannot happen; every (small) deformation of S-(r) and S-[r] is induced naturally by a deformation of S (in particular, all deformations of S-(r) are locally trivial).
