Algebra & Number Theory

Symmetric obstruction theories and Hilbert schemes of points on threefolds

Kai Behrend and Barbara Fantechi

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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 -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 -action and that admits a symmetric obstruction theory compatible with the -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.

Article information

Algebra Number Theory, Volume 2, Number 3 (2008), 313-345.

Received: 4 October 2007
Accepted: 5 November 2007
First available in Project Euclid: 20 December 2017

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Primary: 00A05: General mathematics

symmetric obstruction theories Hilbert schemes Calabi–Yau threefolds $C^*$ actions $S^1$ actions Donaldson–Thomas invariants MNOP conjecture


Behrend, Kai; Fantechi, Barbara. Symmetric obstruction theories and Hilbert schemes of points on threefolds. Algebra Number Theory 2 (2008), no. 3, 313--345. doi:10.2140/ant.2008.2.313.

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