Hodge modules on Shimura varieties and their higher direct images in the Baily–Borel compactification

Abstract

We study the degeneration in the Baily–Borel compactification of variations of Hodge structure on Shimura varieties. Our main result, Theorem 2.6, expresses the degeneration of variations given by algebraic representations in terms of Hochschild, and abstract group cohomology. It is the Hodge theoretic analogue of Pink's theorem on degeneration of étale and ℓ-adic sheaves [Math. Ann. 292 (1992) 197], and completes results by Harder and Looijenga–Rapoport [Eisenstein-Kohomologie arithmetischer Gruppen: Allgemeine Aspekte, Preprint, 1986; Proc. of Symp. in Pure Math., vol. 53, 1991, pp. 223–260]. The induced formula on the level of singular cohomology is equivalent to the theorem of Harris–Zucker on the Hodge structure of deleted neighbourhood cohomology of strata in toroidal compactifications [Inv. Math. 116 (1994) 243].