Higgs bundles, the Toledo invariant and the Cayley correspondence

摘要

Motivated by the study of the topology of the character variety for a non-compact Lie group of Hermitian type G, we undertake a uniform approach, independent of classification theory of Lie groups, to the study of the moduli space of G-Higgs bundles over a compact Riemann surface. We give an intrinsic definition of the Toledo invariant of a G-Higgs bundle which relies on the Jordan algebra structure of the isotropy representation for groups defining a symmetric space of tube type, and prove a general Milnor-Wood type bound of this invariant when the G-Higgs bundle is semistable. Finally, we prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when G is of tube type, which reveals new topological invariants only seen in particular cases from the character variety viewpoint.