Chow-Kunneth decomposition for universal families over Picard modular surfaces, by Andrea Miller, Stefan Muller-Stach, Sigrid Wortmann, Yi-Hu Yang, and Kang Zuo

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.


Andrea Miller <miller@mathi.uni-heidelberg.de>
Stefan Muller-Stach <mueller-stach@mathematik.uni-mainz.de>
Sigrid Wortmann <wortmann@mathi.uni-heidelberg.de>
Yi-Hu Yang <yhyang@mail.tongji.edu.cn>
Kang Zuo <kzuo@mathematik.uni-mainz.de>