The local motivic DT/PT correspondence

We show that the Quot scheme [Formula: see text] parameterising length [Formula: see text] quotients of the ideal sheaf of a line in [Formula: see text] is a global critical locus, and calculate the resulting motivic partition function (varying [Formula: see text]), in the ring of relative motives over the configuration space of points in [Formula: see text]. As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot scheme of [Formula: see text] points of the ideal sheaf [Formula: see text] , where [Formula: see text] is a smooth curve embedded in a smooth 3‐fold [Formula: see text] , 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 [Formula: see text] and of the symmetric product [Formula: see text]. Our ‘relative’ analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map [Formula: see text] , and connections with cohomological Hall algebra representations.