The local motivic DT/PT correspondence

Ben Davison, Andrea T. Ricolfi

Research output: Contribution to journalArticlepeer-review

Abstract

We show that the Quot scheme QnL=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 QnL→Symn(A3), and connections with cohomological Hall algebra representations.
Original languageEnglish
Number of pages51
JournalJournal of the London Mathematical Society
Publication statusAccepted/In press - 9 Apr 2021

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