Hodge modules on Shimura varieties, and their degeneration in the Baily-Borel compactification
type: Item
instance of: scholarly article
title: Hodge modules on Shimura varieties, and their degeneration in the Baily-Borel compactification
zbMATH Open document ID: 1051.14026
DOI: 10.1016/S1631-073X(02)00018-3
author: Jörg Wildeshaus
author: José Ignacio Burgos
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-01-01T00:00:00Z
review text: Let \(j: M \to M^*\) be the Baily-Borel compactification of a Shimura variety \(M\) associated to a reductive group \(G\) over \(\mathbb Q\). Then the boundary \(M^*\setminus M\) has a natural stratification, and each stratum \(i: M_1 \hookrightarrow M^*\) is itself a Shimura variety. On the other hand, any algebraic representation \(\mathbb V\) of \(G\) determines a Hodge module \(\mu (\mathbb V)\) on \(M\). The group \(G_1\) belonging to \(M_1\) is the maximal reductive quotient \(P_1/W_1\) of a certain parabolic subgroup \(P_1\) of \(G\), and the topological inertia group of \(M_1\) in \(M\) is an extension of an arithmetic group \(\overline{H}_C\) by a lattice in \(W_1 (\mathbb Q)\). In this paper the authors express \(i^* j_* \circ \mu\) as a composition of Hochschild cohomology of \(W_1\), abstract cohomology of \(\overline{H}_C\), and the canonical construction on \(M_1\). This result is the Hodge-theoretic analogue of the theorem on \(\ell\)-adic sheaves of \textit{R.~Pink} [Math. Ann. 292, 197-240 (1992; Zbl 0748.14008)].
reviewed by: Min Ho Lee
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 13D03
zbMATH DE Number: 1907818
zbMATH Keywords: Shimura varieties
zbMATH Keywords: Hodge structures
zbMATH Keywords: group cohomology
zbMATH Keywords: Hochschild cohomology
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1907818