The Toledo invariant on smooth varieties of general type

TL;DR

This paper introduces a new definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into Hermitian Lie groups, extending classical results and establishing inequalities and characterizations for maximal representations.

Contribution

It generalizes the Toledo invariant to a broader class of varieties and Lie groups, providing new inequalities and characterizations for maximal representations.

Findings

The Toledo invariant satisfies a Milnor-Wood type inequality for rank ≤ 2 Lie groups.

Maximal representations are characterized within this new framework.

The approach extends classical results from complex hyperbolic lattices to general type varieties.

Abstract

We propose a definition of the Toledo invariant for representations of fundamental groups of smooth varieties of general type into semisimple Lie groups of Hermitian type. This definition allows to generalize the results known in the classical case of representations of complex hyperbolic lattices to this new setting: assuming that the rank of the target Lie group is not greater than two, we prove that the Toledo invariant satisfies a Milnor-Wood type inequality and we characterize the corresponding maximal representations.