2022
DOI: 10.48550/arxiv.2205.04729
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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 deformat… Show more

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Cited by 3 publications
(10 citation statements)
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“…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]).…”
Section: A Introductionsupporting
confidence: 80%
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“…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]).…”
Section: A Introductionsupporting
confidence: 80%
“…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].…”
mentioning
confidence: 82%
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“…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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%