The local motivic DT/PT correspondence

Abstract

We show that the Quot scheme $Q_L^n={\text{Quot}}_{{\mathbb{A}}^3}({\mathcal{I}}_L,n)$ parameterising length $n$ quotients of the ideal sheaf of a line in ${\mathbb{A}}^3$ 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 ${\mathbb{A}}^3$. 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 ${\mathcal{I}}_C\subset {\mathcal{O}}_Y$, where $C\subset Y$ 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 ${\text{Quot}}_Y({\mathcal{I}}_C,n)$ and of the symmetric product ${\text{Sym}}^{n}C$. Our "relative" analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert-Chow map $Q_L^n\to {\text{Sym}}^n({\mathbb{A}}^3)$, and connections with cohomological Hall algebra representations.

49 pages, Final Version

References in corpus (4)

• Non-commutative Donaldson-Thomas theory and the conifold
• Cohomological Hall algebras, vertex algebras and instantons
• Higher rank K-theoretic Donaldson-Thomas theory of points
• The DT/PT correspondence for smooth curves

Cited by in corpus (4)

• On the motive of the nested Quot scheme of points on a curve
• Deformed dimensional reduction
• The geometry of double nested Hilbert schemes of points on curves
• The motive of the Hilbert scheme of points in all dimensions