The Milnor fibre of the Pfaffian and the Hilbert scheme of four points on C^3

TL;DR

This paper investigates the Hodge module on the Hilbert scheme of four points in C^3, linking it to Donaldson-Thomas invariants and the Pfaffian singularity, with explicit weight filtration and E-polynomial computations.

Contribution

It provides an explicit description of the weight filtration and E-polynomial of a Hodge module on the Hilbert scheme, connecting singularity theory with enumerative geometry.

Findings

Explicit weight filtration in terms of intersection cohomology

Computed E-polynomial of the Hodge module

Described the singularity as a Pfaffian degeneracy locus

Abstract

We study a natural Hodge module on the Hilbert scheme of four points on affine three-space, which categorifies the Donaldson--Thomas invariant of the Hilbert scheme. We determine the weight filtration on the Hodge module explicitly in terms of intersection cohomology complexes, and compute the E-polynomial of its cohomology. The computations make essential use of a description of the singularity of the Hilbert scheme as the degeneracy locus of the Pfaffian function.