Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory
Mémoires de la Société Mathématique de France, no. 85 (2001) , 122 p.

In this article, third of a series, we complete the verification of the following fact. The nerve spectral sequence for the cohomology of the Borel-Serre boundary of a Shimura variety Sh is a spectral sequence of mixed Hodge–de Rham structures over the field of definition of its canonical model. To achieve that, we develop the machinery of automorphic vector bundles on mixed Shimura varieties, for the latter enter in the boundary of the toroidal compactifications of Sh ; and study the nerve spectral sequence for the automorphic vector bundles and the toroidal boundary. We also extend the technique of averting issues of base-change by taking cohomology with growth conditions. We give and apply formulas for the Hodge gradation of the cohomology of both Sh and its Borel-Serre boundary.

Dans cet article, troisième d’une série, nous terminons la vérification du fait suivant. La suite spectrale « du nerf », qui calcule la cohomologie du bord de la compactification de Borel-Serre d’une variété de Shimura Sh , est une suite spectrale de structures de Hodge-de Rham mixtes sur le corps de définition de son modèle canonique. Pour le faire, nous développons la théorie de fibrés automorphes sur les variétés de Shimura mixtes, car de tels objets figurent dans le bord d’une compactification toroïdale de Sh  ; et nous étudions la suite spectrale « du nerf » pour les fibrés automorphes et le bord toroïdal. En plus, nous généralisons nos résultats antérieurs sur la cohomologie avec conditions de croissance, qui permettent d’éviter les difficultés associées au changement de base. Enfin, nous énonçons et appliquons des formules pour la graduation de Hodge de la cohomologie de Sh et celle du bord de sa compactification de Borel-Serre.

DOI : https://doi.org/10.24033/msmf.398
Classification:  14G35,  11G18,  14C30,  11F75
Keywords: Shimura varieties, automorphic vector bundles, cohomology of arithmetic groups. mixed Hodge structures
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     author = {Harris, Michael and Zucker, Steven},
     title = {Boundary cohomology of~Shimura~varieties,~III: Coherent cohomology on higher-rank boundary strata and applications to~Hodge~theory},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {85},
     year = {2001},
     doi = {10.24033/msmf.398},
     zbl = {1020.11042},
     mrnumber = {1850830},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2001_2_85__1_0}
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Harris, Michael; Zucker, Steven. Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory. Mémoires de la Société Mathématique de France, Serie 2, , no. 85 (2001), 122 p. doi : 10.24033/msmf.398. http://www.numdam.org/item/MSMF_2001_2_85__1_0/

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