Characteristic forms for holomorphic families of local systems

Abstract

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 $\mathbb{C}[\mathfrak{g}]^G\to \bigoplus_{n\ge 0} H^0(S,\,Ω^n_{S,cl}\otimes R^nf_*\mathbb{C})$, where $Ω^n_{S,cl}$ 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 $S$.