Higher Du Bois and higher rational singularities

Abstract

We prove that the higher direct images R q f * Ω p Y/S of the sheaves of relative Kähler p-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. This is a direct generalization, for the special case of local complete intersections, of a result of Du Bois (case k = 0). As applications, we discuss the behavior of Hodge numbers in families and the unobstructedness of deformations of singular Calabi-Yau varieties.

On k-rational and k-Du Bois local complete intersections

“…This result was also obtained in a different fashion by Friedman and Laza in [FL22b], and previously in the case of isolated singularities in [FL22] (cf. also [FL22c]).…”

“…We also establish some local vanishing results for k-rational and k-Du Bois singularities. Some of these results have been independently obtained in [FL22b].…”

“…In this paper we answer some of these questions positively for local complete intersections, and others, when the V -filtration and minimal exponents are involved, only in the case of hypersurfaces. While writing it, we learned that some of the results we obtain have also been arrived at independently in [FL22b] and [Sai22], following work in the case of isolated singularities in [FL22], [FL22c]; see below. At the moment it seems quite hard to say much beyond the case of local complete intersections.…”

Deformations of Calabi-Yau varieties with $k$-liminal singularities

“…A generalization to the case where Y is allowed to have ordinary double points was proved independently by Kawamata, Ran, and Tian [Kaw92], [Ran92], [Tia92]. Their result was recently further extended in [FL22b] to the case where Y is allowed to have much more general types of singularities (1-Du Bois local complete intersection (lci) singularities in the terminology of Definition 1.4). However, unless we know that there exist smoothings to first order, these unobstructedness results do not directly imply the existence of an actual smoothing.…”

“…k-Du Bois, k rational, and k-liminal singularities. The k-Du Bois and k-rational singularities, natural extensions of Du Bois and rational singularities respectively (case k = 0), were recently introduced by [MOPW21], [JKSY22], [KL20], [FL22b], and [MP22]. The relevance of these classes of singularities (especially for k = 1) to the deformation theory of singular Calabi-Yau and Fano varieties is discussed in [FL22a], which additionally singles out the k-liminal singularities (for k = 1) as particularly relevant to the deformation theory of such varieties.…”