Unipotent flat bundles and Higgs bundles over compact Kähler manifolds

We characterize those unipotent representations of the fundamental group $\pi_1(X,x)$ of a compact Kähler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero.

The characterization is parallel to the one that R. Hain gave of those unipotent representations of $\pi_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle.

We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.