  Let $f:\mathcal{X}\to S$ be a proper holomorphic submersion of complex
manifolds and $G$ a complex reductive linear algebraic group with Lie algebra
$\mathfrak{g}$. Assume also given a holomorphic principal $G$-bundle
$\mathcal{P}$ over $\mathcal{X}$ which is endowed with a holomorphic connection
$\nabla$ relative to $f$ that is flat (this to be thought of as a holomorphic
family of compact complex manifolds endowed with a holomorphic principal
$G$-bundle with flat connection). We show that a refinement of the Chern-Weil
homomorphism yields a graded algebra homomorphism
$\mathbb{C}[\mathfrak{g}]^G\to \bigoplus_{n\ge 0}
H^0(S,\,\Omega^n_{S,cl}\otimes R^nf_*\mathbb{C})$, where $\Omega^n_{S,cl}$
stands for the sheaf of closed holomorphic $n$-forms on $S$. If the fibers of
$f$ are compact Riemann surfaces and we take as our invariant the Killing form,
then we recover Goldman's closed holomorphic $2$-form on the base $S$.
