2022/05/10 by Friedman, Robert, Laza, Radu · 6 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.04729
We prove that the higher direct images Rqf_*Ωp\mathcal Y/S 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≤ 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. This statement has recently been proved for all local complete intersection singularities by Chen-Dirks-Mustaţă.