Graded quiver varieties and derived categories

Abstract

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 δ-functor Φ taking each finite-dimensional representation of 𝒮 to an object of the derived category of Q. We show that the functor Φ establishes a bijection between the strata of the graded affine quiver varieties and the isomorphism classes of objects in the image of Φ. 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.