BOUNDARY COHOMOLOGY OF SHIMURA VARIETIES, III: The nightmare continues

The present article continues the study of the boundary cohomology of Shimura varieties initiated in [HZ1, HZ2]. Let G be a reductive group over Q, X the symmetric space associated to G(R), and Γ an arithmetic subgroup (e.g., a congruence subgroup) of G(Q). We consider the cohomology of Γ\X with coefficients in the local system ̃ V constructed from a representation V of G, i.e., H•(Γ\X, ̃ V) ' H•(Γ, V ). It is standard that this cohomology can be decomposed as the direct sum of “interior” cohomology, defined as the image of the cohomology with compact supports H• c (Γ\X, ̃ V), and a complementary “boundary cohomology” that restricts nontrivially to the boundary of the Borel-Serre (manifold-with-corners) compactification of Γ\X. The designation of boundary cohomology is generally non-canonical, and much work has been devoted to constructing canonical decompositions using Eisenstein series. By an elaboration on the de Rham theorem, one knows that the cohomology group H•(Γ\X, ̃ V) can be expressed as the relative Lie algebra cohomology of the space of V -valued C∞ functions on Γ\G(R), or even the functions of moderate growth ([B2, §7]). Thanks to the work of Franke [Fr1], one can replace the functions of moderate growth by the subspace of automorphic forms, and this can provide the starting point for an approach to the boundary cohomology. However, in this series of articles we are concerned only tangentially with the relation between boundary cohomology and automorphic forms. We choose to work at a more intrinsic level, concentrating instead on the additional structures on H•(Γ\X, ̃ V) when X is a hermitian symmetric domain. In that case, Γ\X is an algebraic variety, and ̃ V underlies a natural variation of Hodge structure. Morihiko Saito’s theory of mixed Hodge modules [Sa3] then gives that H•(Γ\X, ̃ V) has a corresponding mixed Hodge