Nonabelian Hodge correspondence

rdf:langString Nonabelian Hodge correspondence

rdf:langString Nonabelian Hodge theorem

rdf:langString In algebraic geometry and differential geometry, the nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact Kähler manifold. The theorem can be considered a vast generalisation of the Narasimhan–Seshadri theorem which defines a correspondence between stable vector bundles and unitary representations of the fundamental group of a compact Riemann surface. In fact the Narasimhan–Seshadri theorem may be obtained as a special case of the nonabelian Hodge correspondence by setting the Higgs field to zero.

rdf:langString A Higgs bundle has a Hermitian Yang–Mills metric if and only if it is polystable. This metric is a harmonic metric, and therefore arises from a semisimple representation of the fundamental group, if and only if the Chern classes and vanish. Furthermore, a Higgs bundle is stable if and only if it admits an irreducible Hermitian Yang–Mills connection, and therefore comes from an irreducible representation of the fundamental group.

rdf:langString There are homeomorphisms of moduli spaces which restrict to homeomorphisms .

rdf:langString A Higgs bundle arises from a semisimple representation of the fundamental group if and only if it is polystable. Furthermore it arises from an irreducible representation if and only if it is stable.

rdf:langString A representation of the fundamental group is semisimple if and only if the flat vector bundle admits a harmonic metric. Furthermore the representation is irreducible if and only if the flat vector bundle is irreducible.