Smoothing of rational singularities and Hodge structure

We show that the frontier Hodge numbers $h^{p,q}$ (that is, for $pq(n-p)(n-q)=0$) do not change by passing to a desingularization of the singular fiber of a one-parameter degeneration of smooth projective varieties of dimension $n$ if the singular fiber is reduced and has only rational singularities. In this case the order of nilpotence of local monodromy is smaller than the general case by 2, and this does not hold for Du Bois singularities. The proof uses the Hodge filtration of the vanishing cycle Hodge module for the intersection complex of the total space.