Stratifications for Moduli of Sheaves and Moduli of Quiver Representations

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 that the correct parameter space to compare such stratifications is the stack of coherent sheaves, where we construct an asymptotic GIT stratification and prove that this coincides with the Harder–Narasimhan stratification. Then we relate these stratifications for sheaves and quiver representations using a generalisation of a construction of Álvarez-Cónsul and King.