Eichler Integrals in Several Variables

Abstract

In this chapter we study the cohomology of A g . In sections 3, 4, 5 we study the Betti cohomology of A g (ℂ) and its Hodge structure, and show the degeneration of various spectral sequences. Our method, developed in [F 1], is based on the Bernstein-Gelfand-Gelfand (abbreviated as BGG) resolution (cf. [BGG]), Mumford’s extension of equivariant vector bundles to toroidal compactifications (cf. [Mum 6]) and Deligne’s Hodge theory (cf. [D 2], [D 3]). Making use of geometric information available in our present case, we obtain results not contained in [F 1]. In some sense the major work is contained in section 2, where we compute the formal cohomology of toroidal compactifications of fibre products of the universal abelian scheme. In section 6 we describe (but do not present) the p-adic analogue, namely the p-adic etale cohomology groups, which turn out to be crystalline (hence Hodge-Tate) Galois representations and closely related to the crystalline cohomology groups. (Since the crystalline cohomology is closely tied to the de Rham cohomology as is well-known, this analogy is a very good one.) The first section furnishes geometric information needed for studying cohomology, namely explicit compactification of the fibre products of the universal abelian scheme.