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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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