Graded quiver varieties and derived categories

TL;DR

This paper explores the relationship between graded affine quiver varieties and derived categories of quivers, establishing a correspondence via a delta-functor and analyzing the structure of the associated categories.

Contribution

It constructs a delta-functor linking representations of the Nakajima category to the derived category, and characterizes the categories for ADE Dynkin and non-Dynkin quivers.

Findings

The quiver of the Nakajima category S is determined.

A bijection between strata of quiver varieties and derived category objects is established.

For ADE Dynkin quivers, the entire derived category is realized.

Abstract

Inspired by recent work of Hernandez-Leclerc and Leclerc-Plamondon we investigate the link between Nakajima's graded affine quiver varieties associated with an acyclic connected quiver Q and the derived category of Q. As Leclerc-Plamondon have shown, the points of these varieties can be interpreted as representations of a category, which we call the (singular) Nakajima category S. We determine the quiver of S and the number of minimal relations between any two given vertices. We construct a delta-functor Phi taking each finite-dimensional representation of S to an object of the derived category of Q. We show that the functor Phi establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Phi. If the underlying graph of Q is an ADE Dynkin diagram, the image is the whole derived category; otherwise, it is the category of "line bundles over the non commutative curve given by Q". We show that the degeneration order between strata corresponds to Jensen-Su-Zimmermann's degeneration order on objects of the derived category. Moreover, if Q is an ADE Dynkin quiver, the singular category S is weakly Gorenstein of dimension 1 and its derived category of singularities is equivalent to the derived category of Q.