A Bogomolov-Gieseker inequality bounds Chern classes of semistable sheaves by a discriminant inequality. For a slope-semistable torsion-free sheaf
of rank
on a smooth polarized projective complex variety of dimension,
the classical form is
Threefold tilt-stability inequalities also involve the third Chern character. They require additional hypotheses or corrections and are distinct
from the classical inequality.
Bayer, A.; Macrì, E.; and Stellari, P. "The Space of Stability Conditions on Abelian Threefolds, and on Some Calabi-Yau Threefolds."
Invent. Math.206, 869-933, 2016. https://doi.org/10.1007/s00222-016-0665-5.Bayer,
A.; Macrì, E.; and Toda, Y. "Bridgeland Stability Conditions on Threefolds
I: Bogomolov-Gieseker Type Inequalities." J. Alg. Geom.23, 117-163,
2014. https://doi.org/10.1090/S1056-3911-2013-00617-7.
of rank 
on a smooth polarized projective complex variety of dimension 
,
the classical form is
