Higher Du Bois singularities of hypersurfaces

Abstract

For a complex algebraic variety 𝑋, we introduce higher 𝑝-Du Bois singularity by imposing canonical isomorphisms between the sheaves of Kähler differential forms Ω 𝑞 𝑋 and the shifted graded pieces of the Du Bois complex Ω 𝑞 𝑋 for 𝑞 ⩽ 𝑝. If 𝑋 is a reduced hypersurface, we show that higher 𝑝-Du Bois singularity coincides with higher 𝑝-log canonical singularity, generalizing a well-known M S C 2 0 2 0 14B05, 14F10, 14F17, 32S35 (primary)

Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension

“…We also obtain local vanishing results for sheaves of forms with log poles associated to such a resolution, generalising Nakano-type results in [52] and [36]. We prove a vanishing result for cohomologies of the graded pieces of the Du Bois complex when Z is a local complete intersection, extending the study of higher Du Bois singularities of hypersurfaces in [41] and [24]. When Z has isolated singularities, we refine a result in [25] on the coincidence of h-differentials and reflexive differentials, for forms of low degree.…”

“…The first result concerning varieties with this property was obtained in [41], where is was shown that if Z is a hypersurface whose minimal exponent is ≥ 𝑝 + 1, then Z has only higher p-Du Bois singularities; recall that the minimal exponent of Z, which can be defined via the Bernstein-Sato polynomial of Z, roughly describes how close the Hodge filtration and pole order filtration are on the localisation 𝒪 𝑋 ( * 𝑍). The converse to this result was obtained in [24]. The Hodge filtration on local cohomology allows us to extend these results to all local complete intersections.…”

“…We thank S.-J. Jung, I.-K. Kim, M. Saito and Y. Yoon for sharing with us an early version of [24]. We also thank B. Bhatt, L. Ma, M. Saito, K. Schwede and J. Witaszek for their comments on a preliminary version of this paper.…”

“…The following is our main result, relating 𝑝(𝑍) to the behavior of the Du Bois complex of Z. The proof builds on the case of hypersurfaces, which, as already mentioned, is treated in [41] and [24].…”

“…Following the terminology from [24], we say that Z has only higher p-Du Bois singularities if the canonical morphisms Ω 𝑘 𝑍 → Ω 𝑘 𝑍 are isomorphisms for all 0 ≤ 𝑘 ≤ 𝑝. The first result concerning varieties with this property was obtained in [41], where is was shown that if Z is a hypersurface whose minimal exponent is ≥ 𝑝 + 1, then Z has only higher p-Du Bois singularities; recall that the minimal exponent of Z, which can be defined via the Bernstein-Sato polynomial of Z, roughly describes how close the Hodge filtration and pole order filtration are on the localisation 𝒪 𝑋 ( * 𝑍).…”