Quelques relations entre les cohomologies des variétés de Shimura et celles de Griffiths–Schmid (cas du groupe SU(2,1))
Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.
Key Words: Dolbeault cohomologie; automorphic form; unitary groups; Picard variety.