Speaker: Marc Levine
Date: 10:30 13/10/2023
Title: Quadratic Donaldson-Thomas invariants for spin threefolds.

Abstract: There is a general theory of ``motivic’' virtual fundamental classes associated to a perfect 
obstruction theory P on a k-scheme Z, where the virtual fundamental class takes values in the suitably 
twisted Borel-Moore homology of a given motivic cohomology theory E. For Z proper over k, one would 
like to translate this into a ``DT invariant”, living in E^{0,0}(k). For example, if E represents the 
Chow ring or algebraic K-theory, then E_{0,0}^{BM}(k)=\Z and one recovers the usual DT invariant. 
If E is for instance hermitian K-theory, then E^{0,0}(k)=GW(k), the Grothendieck-Witt ring of quadratic 
forms over k, and one would get a quadratic refinement of the usual DT invariant.

For the classical case, one needs the virtual rank  of P to be 0. In the quadratic case, 
one needs in addition a so-called relative orientation of the perfect obstruction theory, this being a 
choice of an invertible sheaf L on Z, and an isomorphism of  det P with L^{\otimes 2}.  Our main result 
is that for X a smooth projective threefold and P the Thomas perfect obstruction theory on Hilb^n(X), 
P admits a relative orientation if X has a spin structure, that is, an isomorphism of the canonical 
sheaf K_X with the square of some invertible sheaf on X. This may be viewed as a refinement of a 
special case of results of Joyce-Upmeier.

We conclude with discussing the example X=P^3. Here Anneloes Viergever has computed the signature of 
the quadratic  DT invariants for Hilb^n(P^3) for n=2,4,6 (the DT invariants has vanishing signature for odd n). 
These agree with the first three coefficients of a power series built out of the MacMahon function.