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We explain how the generalized Milnor-Wood inequality for reductiverepresentations of a cocompact complex-hyperbolic lattice into a Hermitian Liegroup translates, under the non-abelian Hodge correspondence, into variouskinds of Milnor-Wood inequalities for Higgs bundles. This clarifies therelation between the representation theoretic generalized Milnor-Woodinequality and the various different versions of Milnor-Wood inequalities forHiggs bundles that are known in the literature.

Bounded cohomology, Higgs bundles, and Milnor-Wood inequalities