The nonabelian Hodge correspondence relates suitable polystable Higgs bundles on a compact Kähler manifold to semisimple flat bundles, using a
harmonic metric. In the degree-zero curve setting, with the relevant Chern classes
zero, the flat bundles correspond to semisimple representations of the fundamental
group. It identifies the Higgs moduli space and
character variety by a homeomorphism,
rather than generally by an algebraic isomorphism
(Simpson 1994). The coprime nonzero-degree fixed-determinant curve version uses a
twisted character variety with central monodromy
.
Transport through this correspondence is essential in the P=W conjecture, since the two filtrations come from different algebraic structures on the same underlying cohomology.
