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21 Febbraio, 2024 17:00
Seminario Matematico e Fisico di Milano

DT invariants of smooth projective threefolds: classical, quadratic and real

Marc Levine, Universität Duisburg-Essen
Sala di Rappresentanza del Dipartimento di Matematica di via Saldini

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 nth 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.