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

TL;DR

This paper extends the Toledo invariant concept to singular klt varieties of general type, establishing a Milnor-Wood inequality for certain representations and characterizing maximal cases, thus advancing the understanding of their fundamental groups.

Contribution

It generalizes the Toledo invariant to singular varieties and proves a Milnor-Wood inequality for rank ≤ 2 target Lie groups, characterizing maximal representations.

Findings

The Toledo invariant satisfies a Milnor-Wood type inequality for klt varieties.

Maximal representations are characterized in this context.

The work extends classical results to singular algebraic varieties.

Abstract

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.