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Boundary cohomology of Shimura varieties. II: Hodge theory at the boundary. (English) Zbl 0860.11031
[For part I, see the preceding review.]
Let \(G\) be a semisimple algebraic group defined over \(\mathbb{Q}\), \(\Gamma\) a torsion-free, neat arithmetic subgroup, and \(K\) a maximal compact subgroup of \(G(\mathbb{R})\) such that the symmetric space \(G(\mathbb{R})/K\) has a \(G(\mathbb{R})\)-invariant complex structure. An algebraic representation \(G\to GL(V)\) defined over \(\mathbb{Q}\) determines a locally constant sheaf \(\widetilde{\mathbf V}\) over the Shimura variety \(M= \Gamma \backslash G(\mathbb{R})/K\). Assume that \(G\) is of \(\mathbb{Q}\)-rank nonzero, so that \(M\) is noncompact. If \(\widetilde M\) is a toroidal compactification and if \(\partial\widetilde M\) is its boundary, \(\widetilde {\mathbf V}\) can be extended to \(\widetilde M\) and there is a long exact cohomology sequence \[ \cdots \to H^i_c (M,\widetilde {\mathbf V}) \to H^i(M, \widetilde {\mathbf V}) \to H^i (\partial \widetilde M, \widetilde {\mathbf V}) \to\cdots. \] The boundary \(\partial \widetilde M\) of \(\widetilde M\) has a closed covering whose strata correspond to the \(\Gamma \)-conjugacy classes of proper parabolic subgroups of \(G\). Then the nerve of this covering determines the nerve spectral sequence \(\{E_r\}\) which abuts to \(H^\bullet (\partial \widetilde M, \widetilde {\mathbf V})\). The locally constant sheaf \(\widetilde {\mathbf V}\) underlies a polarizable variation of \(\mathbb{Q}\)-Hodge structure, and each term in the long exact cohomology sequence above has a canonical mixed Hodge structure. Furthermore, each mapping in that sequence is a morphism of mixed Hodge structure. In this paper, the authors prove that the nerve spectral sequence is a spectral sequence of mixed Hodge structures. They also determine the mixed Hodge structure on the \(E_1\)-term of the nerve spectral sequence. (Same review submitted to MR).

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
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