Local Vanishing and Hodge Filtration for Rational singularities

Abstract

Given an n-dimensional variety Z with rational singularities, we conjecture that if f : Y → Z is a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, thenWe prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization OX ( * D) is at most n − 3.

Hodge theory of degenerations, (II): vanishing cohomology and geometric applications

“…Finally, (iii) is [Var81a,§4], and (vii) follows from [MP20a, Thm. A] (see also [Sai09] and [MP19,MOP20]). 19 Recall in particular from §2 that σ min f denotes the smallest element of |σ f |.…”

“…In §4, we return to the other extreme: log-canonical singularities, and the even milder klog-canonical singularities with k ≥ 1 (k = 0 being the log-canonical case). This is inspired by the recent work of Mustata-Popa on Hodge ideals [MP19, MP18, MP20b, MP20a, MOP20,Pop18], and some recent work of M. Saito and his collaborators [Sai16,JKYS19a]. After delineating the relationships between the various birational and Hodge-theoretic invariants of singularities (log-canonical threshold, jumping numbers, generation level, period exponent, etc.)…”

𝒟-Modules in Birational Geometry

“…Another line of results proved in [MP16a] and [MOP17] regards the complexity of the Hodge filtration. According to [Sai09], one says that the filtration on a D-module…”

Hodge filtration, minimal exponent, and local vanishing

“…If n ≥ 3 and the divisor D has rational singularities, then the Hodge filtration on O X ( * D) is generated at level n − 3. 1 This was proved when D has isolated singularities, and conjectured to be true in general, in [MOP17]. The general conjecture was already verified recently by Kebekus-Schnell [KS18, §1.3], as a consequence of a local vanishing conjecture; more on this below.…”

“…When i ≥ n − 1 this is shown by elementary methods in [MP16, Theorem B], leading to the coarse bound n − 2 for the generation level of the Hodge filtration mentioned above. When D has rational singularities and i = n − 2, it is proved in [MOP17] in the isolated singularities case, and can be deduced in general from a vanishing statement obtained by Kebekus-Schnell [KS18, Theorem 1.9], which answers [MOP17, Conjecture A]. Using Corollary C, we can in fact obtain a strengthening of this conjecture/statement in the absolute case of a reduced singular hypersurface: by this here we mean a singular complex scheme D, reduced but not necessarily irreducible, that can be embedded as a hypersurface in a smooth variety.…”