  We generalise Simpson's nonabelian Hodge correspondence to the context of
projective varieties with klt singularities. The proof relies on a descent
theorem for numerically flat vector bundles along birational morphisms. In its
simplest form, this theorem asserts that given any klt variety X and any
resolution of singularities, then any vector bundle on the resolution that
appears to come from X numerically, does indeed come from X. Furthermore and of
independent interest, a new restriction theorem for semistable Higgs sheaves
defined on the smooth locus of a normal, projective variety is established.
