Orateur
Description
Let $X$ be a smooth projective 3-fold over the complex numbers. Following work of Thomas, Behrend-Fantechi, and others, one has a virtual fundamental class in the Chow group of 0-cycles on the Hilbert scheme of dimension 0, length $n$ subschemes of $X$, the degree of which is the $n$th Donaldson-Thomas invariant of $X$. Now take $X$ over an arbitrary field $k$. We have developed a construction of virtual fundamental classes with values in an arbitrary motivic cohomology theory. An example of such, a "quadratic" analog of the Chow groups, is the cohomology of the sheaf of Witt rings, which leads to a refinement of the classical DT-invariants to quadratic DT-invariants with values in the Witt ring of quadratic forms over $k$. We will discuss some developments and conjectures for these refined DT invariants, including some computations of the signature of these invariants due to Anneloes Viergever.
