The Du Bois complex of a hypersurface and the minimal exponent

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.…”

“…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].…”

Higher Du Bois singularities of hypersurfaces

“…2.2] (using (12) below and GAGA). We thus get the following (which seems to be proved in [MOPW21] using a slightly different method).…”

“…(This follows from (11-12) below.) The implication (a) ⇒ (b) has been shown in [MOPW21] quite recently. The argument there seems rather complicated calculating the depth of the two sheaves.…”

“…Remark 1.5a. If X is embeddable into a smooth variety Y , it is rather easy to show a noncanonical isomorphism in (1.5.4) as follows (compare to [MOPW21]). Taking an embedded resolution π : (1.5.6) (1.5.7)…”