Updated version received June 14, 2005.
Original version received May 7, 2005.
We discuss the existence of an absolute Chow-Kunneth decomposition for
complete degenerations of families of Abelian threefolds with complex
multiplication over a particular Picard Modular Surface studied by Holzapfel.
In addition to the work of Gordon, Hanamura and Murre we use Relatively
Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular
Surfaces in order to describe complete degenerations of families of abelian
varieties. We furthermore prove vanishing results for cohomology groups of
irreducible representations of certain arithmetic subgroups in SU(2,1) using
the non--compact Simpson type correspondence between the L^2-Higgs
cohomology of the underlying VHS and the L^2-de Rham cohomology resp.
intersection cohomology of local systems.