  Characteristic class relations in Dolbeault cohomology follow from the
existence of a holomorphic Cartan geometry (for example, a holomorphic
conformal structure or a holomorphic projective connection). These relations
can be calculated directly from the representation theory of the structure
group, without selecting any metric or connection or having any knowledge of
the Dolbeault cohomology groups of the manifold. This paper improves on its
predecessor by allowing noncompact and non-K\"ahler manifolds and by deriving
invariants in cohomology of vector bundles, not just in scalar Dolbeault
cohomology, and computing relations involving Chern--Simons invariants in
Dolbeault cohomology. For the geometric structures previously considered in its
predecessor, this paper gives stronger results and simplifies the computations.
It gives the first results on Chern--Simons invariants of Cartan geometries.
