The local motivic DT/PT correspondence

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