Abstract

We prove existence results for Chow–Künneth projectors on compactified universal families of Abelian threefolds with complex multiplication over a particular Picard modular surface studied by Holzapfel. Our method builds up on the approach of Gordon, Hanamura and Murre in the case of Hilbert modular varieties. In addition we use relatively complete models in the sense of Mumford, Faltings and Chai and prove vanishing results for L2–Higgs cohomology groups of certain arithmetic subgroups in SU(2, 1) which are not cocompact.

Introduction

In this paper we discuss conditions for the existence of absolute Chow-Künneth decompositions for families over Picard modular surfaces and prove some partial existence results. In this way we show how the methods of Gordon, Hanamura and Murre [12] can be slightly extended to some cases but fail in some other interesting cases. Let us first introduce the circle of ideas which are behind Chow-Künneth decompositions. For a general reference we would like to encourage the reader to look into [26] which gives a beautiful introduction to the subject and explains all notions we are using.