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
,
where
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
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
points is, up to sign, equal to the Euler characteristic. This proves a conjecture of
Maulik, Nekrasov, Okounkov and Pandharipande.
