Symmetric obstruction theories and Hilbert schemes of points on threefolds

Abstract

In an earlier paper by one of us (Behrend), Donaldson–Thomas type invariants were expressed as certain weighted Euler characteristics of the moduli space. The Euler characteristic is weighted by a certain canonical $\mathbb{Z}$-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point.

Here we evaluate this invariant for the case of a singularity that is an isolated point of a ${ℂ}^{\ast }$-action and that admits a symmetric obstruction theory compatible with the ${ℂ}^{\ast }$-action. The answer is ${\left(-1\right)}^d$, where $d$ is the dimension of the Zariski tangent space.

We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of $n$ points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi–Yau threefold, we deduce that the Donaldson–Thomas invariant of the Hilbert scheme of $n$ points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik, Nekrasov, Okounkov and Pandharipande.