The local motivic DT/PT correspondence

Abstract

We show that the Quot scheme $Q_L^n={Quot}_{A^3}(I_L,n)$ parameterising length $n$ quotients of the ideal sheaf of a line in $A^3$ is a global critical locus, and calculate the resulting motivic partition function (varying $n$ ), in the ring of relative motives over the configuration space of points in $A^3$ . As in the work of Behrend-Bryan-Szendrői, this enables us to define a virtual motive for the Quot scheme of $n$ points of the ideal sheaf $I_C\subset O_Y$ , where $C\subset Y$ is a smooth curve embedded in a smooth 3-fold $Y$ , and we compute the associated motivic partition function. The result fits into a motivic wall-crossing type formula, refining the relation between Behrend's virtual Euler characteristic of ${Quot}_Y(I_C,n)$ and of the symmetric product ${Sym}^{n}C$ . Our 'relative' analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert-Chow map $Q_L^n\to {Sym}^n\left(A^3\right)$ , and connections with cohomological Hall algebra representations.

Keywords: 14C05 (secondary); 14N35 (primary).