Generalized quiver varieties and triangulated categories

TL;DR

This paper introduces generalized quiver varieties that unify classical and cyclic cases, exploring their geometry through algebra P and relating their strata to triangulated categories, with applications to type A quiver varieties.

Contribution

It defines generalized quiver varieties, links their geometry to algebra P, and establishes a correspondence with triangulated categories, extending classical results.

Findings

Strata of generalized quiver varieties correspond to objects in proj P.

Degeneration order matches Jensen-Su-Zimmermann's order.

Type A quiver varieties are realized as moduli spaces of algebra S.

Abstract

In this paper, we introduce generalized quiver varieties which include as special cases classical and cyclic quiver varieties. The geometry of generalized quiver varieties is governed by a finitely generated algebra P: the algebra P is self-injective if the quiver Q is of Dynkin type, and coincides with the preprojective algebra in the case of classical quiver varieties. We show that in the Dynkin case the strata of generalized quiver varieties are in bijection with the isomorphism classes of objects in proj P, and that their degeneration order coincides with the Jensen-Su-Zimmermann's degeneration order on the triangulated category proj P. Furthermore, we prove that classical quiver varieties of type A can be realized as moduli spaces of representations of an algebra S.