Degenerations in graded quiver varieties and in derived categories of Dynkin quivers

TL;DR

This paper explores the relationship between degenerations in graded quiver varieties and derived categories of Dynkin quivers, establishing a correspondence that enhances understanding of their structural interplay.

Contribution

It demonstrates that all degenerations in the derived category of a Dynkin quiver can be realized through stratifications in graded quiver varieties, extending previous results.

Findings

Degenerations in derived categories correspond to strata degenerations in quiver varieties.

All derived category degenerations can be obtained via stratification methods.

The partial order of Jensen-Su-Zimmermann is fully characterized for Dynkin quivers.

Abstract

For any acyclic quiver, Keller-Scherotzke provided a stratifying functor from the category of finite-dimensional modules of the singular Nakajima category to the derived category of the quiver. Under this functor, a degeneration of strata of a graded quiver variety corresponds to a degeneration, in the sense of Jensen-Su-Zimmermann, in the derived category. In this article, for any Dynkin quiver, we further investigate Jensen-Su-Zimmermann's partial order and show that any degeneration of objects in the derived category can be obtained in this way.