Milnor-Wood type inequalities for Higgs bundles

We explain how the generalized Milnor–Wood inequality of Burger and Iozzi for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates under the non-abelian Hodge correspondence into an inequality for topological invariants of the corresponding Higgs bundles. In this manner, we obtain in a uniform way a universal Milnor–Wood inequality for Higgs bundles over complex-hyperbolic manifolds of arbitrary dimensions and with arbitrary Hermitian structure group. This complements results of Biquard, Bradlow, García-Prada, Gothen, Mundet, Rubio and Chaput, Koziarz, Maubon.