Portal de Periódicos da CAPES

Symmetric obstruction theories and Hilbert schemes of points on threefolds

2008; Mathematical Sciences Publishers; Volume: 2; Issue: 3 Linguagem: Inglês

10.2140/ant.2008.2.313

1944-7833

Kai Behrend, Barbara Fantechi,

Algebraic structures and combinatorial models

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