Translation quiver varieties

TL;DR

This paper introduces translation quiver varieties, a unifying framework that generalizes Nakajima quiver varieties, and demonstrates their geometric properties and computational methods for motivic classes.

Contribution

It defines translation quiver varieties, shows their fixed points are also translation quiver varieties, and provides an algorithm to compute their motivic classes.

Findings

Translation quiver varieties are smooth and pure.

They have Tate motivic classes.

An algorithm to compute motivic classes is provided.

Abstract

We introduce a framework of translation quiver varieties which includes Nakajima quiver varieties as well as their graded and cyclic versions. An important feature of translation quiver varieties is that the sets of their fixed points under toric actions can be again realized as translation quiver varieties. This allows one to simplify quiver varieties in several steps. We prove that translation quiver varieties are smooth, pure and have Tate motivic classes. We also describe an algorithm to compute those motivic classes.