Motivic degree zero Donaldson-Thomas invariants

TL;DR

This paper introduces a motivic refinement of degree zero Donaldson-Thomas invariants for threefolds, providing explicit formulas and computations for affine space and general cases, connecting to MacMahon functions.

Contribution

It defines the virtual motive of Hilbert schemes on threefolds and computes their generating functions using motivic techniques, extending known results to higher dimensions.

Findings

Virtual motives of Hilbert schemes are computed for affine three-space.

The generating function is expressed as a motivic exponential.

A product formula involving deformed MacMahon functions is derived.

Abstract

Given a smooth complex threefold X, we define the virtual motive of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, this gives a motivic refinement of the n-point degree zero Donaldson-Thomas invariant of X. The key example is affine three-space, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef-Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives of the Hilbert schemes of affine three-space via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Gottsche's formula for the Poincare polynomials of the Hilbert schemes of points on surfaces.