Higher Du Bois and higher rational singularities

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}.