01518nas a2200109   4500008004300000245007900043210006900122520114200191100001701333700002201350856003601372       2008                        en_Ud  00aSymmetric obstruction theories and Hilbert schemes of points on threefolds0 aSymmetric obstruction theories and Hilbert schemes of points on 3 aIn 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\\nZ-valued constructible function on the moduli space. This constructible function associates to\\nany 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\\nis 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.1 aBehrend, Kai1 aFantechi, Barbara  uhttp://hdl.handle.net/1963/270900729nas a2200121   4500008004300000245002700043210002700070520041600097100002200513700001800535700001800553856003600571       2007                        en_Ud  00aSmooth toric DM stacks0 aSmooth toric DM stacks3 aWe give a new definition of smooth toric DM stacks in the same spirit of toric varieties. We show that our definition is equivalent to the one of Borisov, Chen and Smith in terms of stacky fans. In particular, we give a geometric interpretation of the combinatorial data contained in a stacky fan. We also give a bottom up classification in terms of simplicial toric varieties and fiber products of root stacks.1 aFantechi, Barbara1 aMann, Etienne1 aNironi, Fabio  uhttp://hdl.handle.net/1963/2120