  We review and apply Cheeger-Gromov theory on $l^2$-cohomology of infinite
coverings of complete manifolds with bounded curvature and finite volume.
Applications focus on $l^2$-cohomology of (pullback of) harmonic Higgs bundles
on some covering of Zariski open sets of K\"ahler manifolds. The
$l^2-$Dolbeault to DeRham spectral sequence of these Higgs bundles is seen to
degenerate at $E_2$.
