A Harder-Narasimhan filtration is a finite filtration whose nonzero successive factors are semistable with strictly decreasing slopes or phases. For a coherent sheaf, the slope depends on the chosen polarization and may include a divisor twist. For a stability condition on a triangulated category, successive distinguished triangles replace short exact sequences, and the factors have decreasing real phases.
For a torsion-free sheaf of positive rank on a polarized threefold, the twisted slope is
 |
Here 
is ample and 
is a real numerical divisor class. A nonzero torsion sheaf
is assigned slope 
in this convention. Semistability compares the slopes
of subobjects and quotients; existence of the filtration is a separate property of
the chosen stability framework.
