#### Vol. 2, No. 3, 2008

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Symmetric obstruction theories and Hilbert schemes of points on threefolds

### Kai Behrend and Barbara Fantechi

Vol. 2 (2008), No. 3, 313–345
##### 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 ${ℂ}^{\ast }$-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.

##### Keywords
symmetric obstruction theories, Hilbert schemes, Calabi–Yau threefolds, $C^*$ actions, $S^1$ actions, Donaldson–Thomas invariants, MNOP conjecture
Primary: 00A05