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 theorem for p = 0. The assertion that p-log canonicity implies p-Du Bois has been proved by Mustata, Olano, Popa, and Witaszek quite recently calculating the depth of the two sheaves. Our method seems much simpler using directly the acyclicity of Koszul complex in a certain range, which enables us to produce the desired isomorphisms immediately. We also improve some non-vanishing assertion shown by them, using the Tjurina subspectrum in the isolated singularity case. This is useful for instance to estimate the lower bound of the maximal root of the reduced Bernstein-Sato polynomial in the case where a quotient singularity is a hypersurface and its singular locus has codimension at most 4.

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, generalizing Nakano-type results in [Sai07] and [MP19a]. 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 [MOPW21] and [JKSY21]. When Z has isolated singularities, we refine a result in [KS21] on the coincidence of h-differentials and reflexive differentials, for forms of low degree.…”

“…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 [MOPW21] and [JKSY21].…”

“…Following the terminology from [JKSY21], 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 [MOPW21], 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 localization O X ( * Z). The converse to this result was obtained in [JKSY21]. The Hodge filtration on local cohomology allows us to extend these results to all local complete intersections.…”

“…Proof of Theorem F. We follow the approach in [MOPW21] and [JKSY21], which treat the two implications in the theorem for hypersurfaces. We may and will assume that Z is singular, since otherwise the equivalence is trivial.…”

“…We thank S.-J. Jung, I.-K. Kim, M. Saito, and Y. Yoon for sharing with us an early version of [JKSY21]. We also thank B. Bhatt, L. Ma, M. Saito, and K. Schwede for their comments on a preliminary version of this paper.…”