The local motivic DT/PT correspondence

We show that the Quot scheme (Formula presented.) parameterising length (Formula presented.) quotients of the ideal sheaf of a line in (Formula presented.) is a global critical locus, and calculate the resulting motivic partition function (varying (Formula presented.)), in the ring of relative motives over the configuration space of points in (Formula presented.). As in the work of Behrend\u2013Bryan\u2013Szendr\u151i, this enables us to define a virtual motive for the Quot scheme of (Formula presented.) points of the ideal sheaf (Formula presented.), where (Formula presented.) is a smooth curve embedded in a smooth 3-fold (Formula presented.), 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 presented.) and of the symmetric product (Formula presented.). Our \u2018relative\u2019 analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert\u2013Chow map (Formula presented.), and connections with cohomological Hall algebra representations