Stratifications of Parameter Spaces for Complexes by Cohomology Types

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.