  We prove that the higher direct images $R^qf_*\Omega^p_{\mathcal Y/S}$ of the
sheaves of relative K\"ahler differentials are locally free and compatible with
arbitrary base change for flat proper families whose fibers have $k$-Du Bois
local complete intersection singularities, for $p\leq k$ and all $q\geq 0$,
generalizing a result of Du Bois (the case $k=0$). We then propose a definition
of $k$-rational singularities extending the definition of rational
singularities, and show that, if $X$ is a $k$-rational variety with either
isolated or local complete intersection singularities, then $X$ is $k$-Du Bois.
As applications, we discuss the behavior of Hodge numbers in families and the
unobstructedness of deformations of singular Calabi-Yau varieties.
  In an appendix, Morihiko Saito proves that, in the case of hypersurface
singularities, the $k$-rationality definition proposed here is equivalent to a
previously given numerical definition for $k$-rational singularities. As an
immediate consequence, it follows that for hypersurface singularities, $k$-Du
Bois singularities are $(k-1)$-rational. This statement has recently been
proved for all local complete intersection singularities by
Chen-Dirks-Musta\c{t}\u{a}.
