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 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^*-action and that admits a symmetric obstruction theory compatible with the C^*-action. The answer is (−1)^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.

Keywords

symmetric obstruction theories, Hilbert schemes, Calabi–Yau threefolds, C* actions, S¹ actions, Donaldson–Thomas invariants, MNOP conjecture

Mathematical Subject Classification

Primary: 00A05

Authors