Hodge Ideals

Abstract

We use methods from birational geometry to study the Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of ideal sheaves, called Hodge ideals, the first of which is a multiplier ideal. We analyze their local and global properties, and use them for applications related to the singularities and Hodge theory of hypersurfaces and their complements.

Hodge filtration, minimal exponent, and local vanishing

“…Since α D > 0, Theorem A recovers in particular the fact that F • O X ( * D) is always generated at level n − 2, proved in [MP16,Theorem B]. Note also that it is possible to do better than Theorem A: as an extreme case, if D is a singular simple normal crossing divisor, then F • O X ( * D) is generated at level 0, but α D = 1.…”

“…In this paper we give a bound for the generation level of the Hodge filtration on D Xmodules naturally associated to rational multiples of a reduced effective divisor D on X, in terms of data provided by the Bernstein-Sato polynomial of D. This study was initiated by Saito [Sai09], who provided such bounds for special types of singularities. Some general results were later found in [MP16], [MP19]. We improve them here, using the main result of [MP18], and also exploit the fact that they are, somewhat surprisingly, related to local vanishing theorems for sheaves of forms with log poles in birational geometry.…”

“…Therefore Theorem E provides an effective bound describing which higher Hodge ideals of αD are fully determined by lower ones. This type of result is very useful for concrete calculations of Hodge ideals, see [MP16] and [MP19].…”

“…See e.g. [MP16] for an in-depth study of this filtration. If D is smooth, then the filtration is generated at level 0, hence from now on we focus on the case when D is singular.…”

“…As a special case, we naturally identify M(f −1 ) with the usual localization O X ( * D). In particular, when D is reduced and α = 1, this gives the Hodge ideals considered in [MP16].…”