Abstract:

Graber and Pandharipande extended the classical Bott residue theorem to the computation of torus-equivariant virtual fundamental classes in terms of the restriction to the fixed points, and their result was used in papers by Maulik, Nekrasov, Okounkov and Pandharipande to compute the dimension zero Donaldson-Thomas invariants for smooth toric threefolds over the complex numbers. We have developed a version of virtual localization for Witt sheaf cohomology, where one needs to replace a \G_m-action with an action by the normalizer of the diagonal torus in SL_2. This was extended to SL[\eta^{-1}]-oriented motivic cohomology theories by Alessadro D’Angelo in his Ph.D. thesis. We will discuss this circle of ideas and describe an application to computing the ``real'' dimension zero Donaldson-Thomas invariants for (\P^1)^3, a joint work with A. Viergever.