On Invariants for Representations of Complex Hyperbolic Lattices

This thesis contributes to the understanding of connected components of character varieties for representations through the study of associated numerical invariants, which are of interest in their own right, using bounded cohomology. We focus on two classical invariants: the Euler number, associated to representations of torsion-free complex hyperbolic lattices into PU(n,1), and the Toledo invariant, defined more generally for representations into simple non-compact Hermitian Lie groups with finite center. Both invariants admit differential-geometric, algebro-geometric, and cohomological interpretations. They serve as powerful tools in establishing rigidity results and play an important role in higher Teichmüller theory. The central objective of this work is to analyze the nature of the values taken by the Euler number and the Toledo invariant. For higher-dimensional lattices, we show that both invariants take integral values (up to rescaling). Furthermore, by proving their continuity, we conclude that they are constant on connected components of the corresponding character varieties. Bounded cohomology plays a crucial role in our approach, as it provides a framework that extends the classical definitions of the invariants from uniform to non-uniform lattices. Among the key contributions of this thesis is the identification of a suitable formulation for the Toledo invariant, understood as the rescaled degree, and placed within a unified setting alongside the bounded-cohomological definition of the Euler number, which was previously introduced and studied as the rescaled volume in the real hyperbolic case by Bucher, Burger and Iozzi. This unified formulation, in turn, allows to establish our integrality results that emerge from an interplay between different cohomology theories and constructions involving several variants of Chern classes. Finally, we introduce new invariants associated with mixed representations, inspired by the study of volumes for lattices in the context of Anti-de-Sitter geometry and its generalizations by Tholozan. For these invariants as well, we show that the integrality phenomenon persists.