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**Boundary cohomology of Shimura varieties. III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory.**
*(English)*
Zbl 1020.11042

For Parts I and II, cf. Ann. Sci. Éc. Norm. Supér. (4) 27, 249-344 (1994; Zbl 0860.11030); Invent. Math. 116, 243-308 (1994; Zbl 0860.11031) and Invent. Math. 121, 437 (1995; Zbl 1008.11529).

Let \(G\) be a reductive group over \(\mathbb Q\), and let \(X\) be the symmetric space associated to \(G(\mathbb R)\). Given a discrete subgroup \(\Gamma\) of \(G(\mathbb Q)\) and a representation \(V\) of \(G\), the group cohomology \(H^\bullet (\Gamma, V)\) can be identified with the cohomology \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) of the locally symmetric space \(\Gamma \setminus X\) with coefficients in the local system \(\widetilde{\mathbb V}\) corresponding to \(V\). The cohomology \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) can be decomposed as the direct sum of the interior cohomology, defined as the image of the cohomology with compact supports, and the boundary cohomology that restricts nontrivially to the boundary of the Borel-Serre compactification of \(\Gamma \setminus X\). The adelic version of \(\Gamma \setminus X\) is the Shimura variety \(\text{Sh} (G,X)\), whose connected components are of the form \(\Gamma \setminus X\). This has a canonical model over a number field \(E\), and \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) can be identified with the hypercohomology of an \(E\)-rational complex of coherent sheaves on \(\text{Sh} (G,X)\). In particular, \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) has a Hodge filtration whose graded pieces are given by the coherent cohomology with coefficients in certain automorphic vector bundles.

In this paper, the authors prove that the nerve spectral sequence of the Borel-Serre boundary of the Shimura variety \(\text{Sh} (G,X)\) is a spectral sequence of mixed Hodge-de Rham structures over \(E\) by developing the machinery of automorphic vector bundles on mixed Shimura varieties and studying the nerve spectral sequence for those bundles and the toroidal boundary. They also extend the technique of averting issues of base change by taking cohomology with growth conditions and provide formulas for the Hodge gradation of the cohomology of both \(\text{Sh} (G,X)\) and its Borel-Serre boundary.

Let \(G\) be a reductive group over \(\mathbb Q\), and let \(X\) be the symmetric space associated to \(G(\mathbb R)\). Given a discrete subgroup \(\Gamma\) of \(G(\mathbb Q)\) and a representation \(V\) of \(G\), the group cohomology \(H^\bullet (\Gamma, V)\) can be identified with the cohomology \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) of the locally symmetric space \(\Gamma \setminus X\) with coefficients in the local system \(\widetilde{\mathbb V}\) corresponding to \(V\). The cohomology \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) can be decomposed as the direct sum of the interior cohomology, defined as the image of the cohomology with compact supports, and the boundary cohomology that restricts nontrivially to the boundary of the Borel-Serre compactification of \(\Gamma \setminus X\). The adelic version of \(\Gamma \setminus X\) is the Shimura variety \(\text{Sh} (G,X)\), whose connected components are of the form \(\Gamma \setminus X\). This has a canonical model over a number field \(E\), and \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) can be identified with the hypercohomology of an \(E\)-rational complex of coherent sheaves on \(\text{Sh} (G,X)\). In particular, \(H^\bullet (\Gamma \setminus X, \widetilde{\mathbb V})\) has a Hodge filtration whose graded pieces are given by the coherent cohomology with coefficients in certain automorphic vector bundles.

In this paper, the authors prove that the nerve spectral sequence of the Borel-Serre boundary of the Shimura variety \(\text{Sh} (G,X)\) is a spectral sequence of mixed Hodge-de Rham structures over \(E\) by developing the machinery of automorphic vector bundles on mixed Shimura varieties and studying the nerve spectral sequence for those bundles and the toroidal boundary. They also extend the technique of averting issues of base change by taking cohomology with growth conditions and provide formulas for the Hodge gradation of the cohomology of both \(\text{Sh} (G,X)\) and its Borel-Serre boundary.

Reviewer: Min Ho Lee (Cedar Falls)

### MSC:

11G18 | Arithmetic aspects of modular and Shimura varieties |

14G35 | Modular and Shimura varieties |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

11F75 | Cohomology of arithmetic groups |