Hodge filtration, minimal exponent, and local vanishing

Abstract

We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are extended to Q-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.1 As mentioned above, for n = 2 the filtration is always generated at level 0.

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

“…. This interpretation and its extension to Q-divisors have proven to be very useful for studying both the Hodge filtration and the minimal exponent, see [MP20a], [MP20b].…”

“…Various bounds on the local cohomological dimension that are relevant to this theorem can be found in §6. At least for local complete intersections, the range of vanishing in Theorem D could perhaps be further improved by analogy with [MP20b,Theorem D], though this will rely on connections with the Bernstein-Sato polynomial of Z not known at the moment; cf. Remark 9.10.…”

Hodge ideals and minimal exponents of ideals

“…In this section we define and study an extension of the concept of minimal exponent of a hypersurface [Sai93], [Sai16] (see also [MP18b], [MP19b] for a recent study and applications) to the case of arbitrary subschemes. As always, we work on a smooth variety X of dimension n.…”

Hodge ideals of free divisors

“…The aim of this paper is to describe the Hodge filtration on the mixed Hodge module j * Q H U [n] for certain divisors and to compute it explicitly in some examples. This question has some history, but has recently been reconsidered in a series of articles (see [28][29][30][31]) by Mustaţȃ and Popa (in the algebraic setting though), from a birational point of view. The authors of these papers introduce the so-called Hodge ideals: these are coherent sheaves of ideals I k (D) ⊂ O X measuring the difference between the Hodge filtration F H…”