Milnor-Wood inequality for klt varieties of general type and uniformization

We generalize the definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type due to Koziarz and Maubon to the context of singular klt varieties, where the natural fundamental groups to consider are those of smooth loci. Assuming that the rank of the target Lie group is not greater than two, we show that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.
Comment: v2: 15 pages, minor changes as requested by referee. Accepted for publication in Bulletin of the London Mathematical Society; v3: compared to the published version, this version fixes some typos and gives a more precise characterization of the situation in which equality is attained in the Milnor-Wood inequality