Higher Du Bois and higher rational singularities

. We prove that the higher direct images R q f ∗ Ω p 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 ≤ k and all q ≥ 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. Independently, we have proved that the latter statement also holds for isolated local complete intersection singularities, and conjecture that it holds more generally for all local complete intersection singularities.