  Moduli spaces of stable objects in the derived category of a $K3$ surface
provide a large class of holomorphic symplectic varieties. In this paper, we
study the interplay between Chern classes of stable objects and zero-cycles on
holomorphic symplectic varieties which arise as moduli spaces.
  First, we show that the second Chern class of any object in the derived
category lies in a suitable piece of O'Grady's filtration on the
$\mathrm{CH}_0$-group of the $K3$ surface. This solves a conjecture of O'Grady
and improves on previous results of Huybrechts, O'Grady, and Voisin. Then we
propose a candidate of the Beauville-Voisin filtration on the
$\mathrm{CH}_0$-group of the moduli space of stable objects. We discuss its
connection with Voisin's recent proposal via constant cycle subvarieties. In
particular, we deduce the existence of algebraic coisotropic subvarieties in
the moduli space.
  Further, for a generic cubic fourfold containing a plane, we establish a
connection between zero-cycles on the Fano variety of lines and on the
associated $K3$ surface.
