The local motivic DT/PT correspondence.

Abstract

We show that the Quot scheme QLn=QuotA3(IL,n) parameterising length n quotients of the ideal sheaf of a line in A3 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 A3. 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 IC⊂OY, where C⊂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 QuotY(IC,n) and of the symmetric product SymnC. Our ‘relative’ analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map QLn→Symn(A3), and connections with cohomological Hall algebra representations.