Stratifications of parameter spaces for complexes by cohomology types

TL;DR

This paper investigates how stability conditions for complexes of sheaves vary with a parameter, revealing that the Harder-Narasimhan filtrations of complexes reflect those of their cohomology sheaves and connecting stratifications by stability and HN types.

Contribution

It establishes a relationship between stability conditions parametrized by a rational parameter and the filtrations of complexes and their cohomology sheaves, linking stratifications by stability and HN types.

Findings

Harder-Narasimhan filtrations of complexes encode those of cohomology sheaves.

Stratification by stability parameters aligns with stratification by HN types.

Small parameter values reveal the filtrations of cohomology sheaves.

Abstract

We study a collection of stability conditions (in the sense of Schmitt) for complexes of sheaves over a smooth complex projective variety indexed by a positive rational parameter. We show that the Harder-Narasimhan filtration of a complex for small values of this parameter encodes the Harder-Narasimhan filtrations of the cohomology sheaves of this complex. Finally we relate a stratification into locally closed subschemes of a parameter space for complexes associated to these stability parameters with the stratification by Harder-Narasimhan types.