Higher Du Bois and higher rational singularities

Abstract

We prove that the higher direct images $R^q{f}_{\ast }{\mathrm{\Omega }}_{\mathcal{Y}∕S}^p$ of the sheaves of relative Kähler 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 $p\le k$ and all $q\ge 0$, 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, and Mustaţă.