Stratifications for moduli of sheaves and moduli of quiver representations

TL;DR

This paper compares stratifications of parameter spaces for sheaves and quiver representations, revealing their differences and connections through geometric invariant theory and stack analysis.

Contribution

It clarifies the relationship between stratifications for sheaves and quiver representations, extending previous constructions to relate their moduli spaces.

Findings

Stratifications coincide for quiver representations.

Stratifications differ for sheaves on Quot schemes.

The stack of coherent sheaves is the correct space for comparison.

Abstract

We study the relationship between two stratifications on parameter spaces for coherent sheaves and for quiver representations: a stratification by Harder-Narasimhan types and a stratification arising from the geometric invariant theory construction of the associated moduli spaces of semistable objects. For quiver representations, both stratifications coincide, but this is not quite true for sheaves. We explain why the stratifications on various Quot schemes do not coincide and see that the correct parameter space to compare such stratifications is the stack of coherent sheaves. Then we relate these stratifications for sheaves and quiver representations using a generalisation of a construction of \'Alvarez-C\'onsul and King.