On the Hodge Filtration of Hodge Modules

Let X be a complex manifold, and Z an irreducible closed analytic subset. We have the polarizable Hodge Module ICZQ H whose underlying perverse sheaf is the intersection complex ICZQ. See [16]. Let (M,F ) be its underlying filtered DX -Module. ThenM is the unique regular holonomic DX -Module which corresponds to ICZC by the Riemann-Hilbert correspondence [9] [13], and it is relatively easy to determineM in some cases (for example, if Z is a hypersurface with isolated singularities [28]). However, the Hodge filtration F on M is a more delicate object, and it is not easy to describe F explicitly, because we have to calculate the filtration V to some extent. More generally, assume (M,F ) underlies a polarizable Hodge Module M with strict support Z (see [16]). Let q = min{p ∈ Z : FpM 6= 0}. The generating level of (M,F ) is defined to be the minimal length of a part of the filtration F which generates (M,F ) over (DX , F ) (i.e., (M,F ) has generating level ≤ r if Fq+r+iM = FiDXFq+rM for i ≥ 0, and generating level r if it has generating level ≤ r but not ≤ r − 1). Here the filtration F on DX is by the order of differential operator. Note that (M,F ) has generating level ≤ r for r ≫ 0 at least locally, because F is a good filtration (i.e., GrM is coherent over GrDX). The notion of generating level measures in some sense the complexity of the filtration F . (See also Remark (ii) after (1.3).) If Z is a point so that M corresponds to a Hodge structure, then the generating level of (M,F ) coincides with the level of the Hodge structure.