Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory

Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory
[Cohomologie au bord des variétés de Shimura, III]

Harris, Michael ; Zucker, Steven

Mémoires de la Société Mathématique de France, no. 85 (2001) , 122 p.

Résumé
Abstract

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.

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.

MR Zbl | 1 citation dans Numdam

DOI : 10.24033/msmf.398

Classification : 14G35, 11G18, 14C30, 11F75
Keywords: Shimura varieties, automorphic vector bundles, cohomology of arithmetic groups. mixed Hodge structures
Mot clés : Variétés de Shimura, fibrés vectoriels automorphes, cohomologie de groupes arithmétiques, structures de Hodge mixtes

@book{MSMF_2001_2_85__1_0,
     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},
     mrnumber = {1850830},
     zbl = {1020.11042},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2001_2_85__1_0/}
}

TY  - BOOK
AU  - Harris, Michael
AU  - Zucker, Steven
TI  - Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory
T3  - Mémoires de la Société Mathématique de France
PY  - 2001
IS  - 85
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2001_2_85__1_0/
DO  - 10.24033/msmf.398
LA  - en
ID  - MSMF_2001_2_85__1_0
ER  -

%0 Book
%A Harris, Michael
%A Zucker, Steven
%T Boundary cohomology of Shimura varieties, III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory
%S Mémoires de la Société Mathématique de France
%D 2001
%N 85
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2001_2_85__1_0/
%R 10.24033/msmf.398
%G en
%F MSMF_2001_2_85__1_0

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, Série 2, no. 85 (2001), 122 p. doi : 10.24033/msmf.398. http://numdam.org/item/MSMF_2001_2_85__1_0/

Bibliographie
Cité par

[AMRT] Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S., Smooth compactification of Locally Symmetric Varieties, Math. Sci. Press, Brookline, MA, 1975. | MR | Zbl

[BHR] Blasius, D., Harris, M., Ramakrishnan, D., Coherent cohomology, limits of discrete series, and Galois conjugation, Duke Math. J., 73, 647–685 (1994). | MR | Zbl

[B1] Borel, A., Introduction to automorphic forms, PSPM 9, In: Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. AMS, 9, 199–210 (1966).

[B2] Borel, A., Stable real cohomology of arithmetic groups, Ann. Scient. Ec. Norm. Sup., 7, 235–272 (1974). | EuDML | Zbl | Numdam

[B3] Borel, A., Introduction aux Groupes Arithmétiques, Hermann, Paris, 1969. | Zbl

[BG] Borel, A., Garland, H., Laplacian and the discrete spectrum of an arithmetic group, Am. J. Math., 105, 309–335 (1983). | Zbl

[BS] Borel, A., Serre, J.-P., Corners and arithmetic groups. Comm. Math. Helv., 48, 436–491 (1973). | EuDML | Zbl

[De1] Deligne, P., Equations différentielles à points singuliers réguliers. Lecture Notes in Math., 163, (1970). | Zbl

[De2] Deligne, P., Théorie de Hodge, III. Publ. Math. IHES 44, 5–77 (1974). | EuDML | Zbl | Numdam

[E] El Zein, F., Mixed Hodge structures, Trans. AMS 275, 71–106 (1983).

[F] Faltings, G., On the cohomology of locally symmetric hermitian spaces, Lect. Notes in Math., 1029, 55-98 (1984).

[Fr1] Franke, J., Harmonic analysis in weighted $L_{2}$ spaces, Ann. Sci. Ec. Norm. Sup., 31, 181-279 (1998). | EuDML | Zbl | Numdam

[Fr2] Franke, J., A topological model for some summand in the Eisenstein cohomology of congruence subgroups (appended to [KR]). | Zbl

[GH] Griffiths, P., Harris, J., Principles of Algebraic Geometry, John Wiley and Sons, New York (1978). | Zbl

[Ha] Harder, G., On the cohomology of discrete arithmetically defined groups. In: Proceedings of the International Colloquium on Discrete Subgroups of Lie Groups and Applications to Moduli. Oxford Univ. Press, Bombay, 1975, 129–160. | Zbl

[H1] Harris, M., Arithmetic vector bundles and automorphic forms on Shimura varieties. I. Invent. Math., 82, 151-189 (1985); II, Compositio Math., 60, 323–378 (1986). | EuDML | Zbl | Numdam

[H2] Harris, M., Functorial properties of toroidal compactifications of locally symmetric varieties, Proc. Lon. Math. Soc. 59, 1–22 (1989). | Zbl

[H3] Harris, M., Automorphic forms and the cohomology of vector bundles on Shimura varieties, in L. Clozel and J.S. Milne, eds., Proceedings of the Conference on Automorphic Forms, Shimura Varieties, and $L$ -functions, Ann Arbor, 1988, Perspectives in Mathematics, New York: Academic Press, Vol. II, 41–91 (1989).

[H4] Harris, M., Automorphic forms of $\bar{\partial}$ -cohomology type as coherent cohomology classes, J. Diff. Geom., 32, 1–63 (1990) | Zbl

[H5] Harris, M., Hodge-de Rham structures and periods of automorphic forms. In: Motives (Seattle, 1991), Proc. Symp. Pure Math. AMS, 55, Part 2, pp. 573–624 (1994). | Zbl

[HP] Harris, M., Phong, D.H., Cohomologie de Dolbeault à croissance logarithmique à l’infini. C. R. Acad. Sci. Paris, 302, 307–310 (1986). | Zbl

[HZ1] Harris, M., Zucker, S.: Boundary cohomology of Shimura varieties, I: coherent cohomology on the toroidal boundary, Ann. Scient. Ec. Norm. Sup., 27, 249–344 (1994). | EuDML | Zbl | Numdam

[HZ2] Harris, M., Zucker, S.: Boundary cohomology of Shimura varieties, II: Hodge theory at the boundary, Inventiones Math., 116, 243–307 (1994); Erratum, Inventiones Math., 121, p. 437 (1995). | EuDML | Zbl

[Ka] Kashiwara, M.: The asymptotic behavior of a variation of polarized Hodge structure, Publ. RIMS Kyoto Univ., 21 (1985), 853–875. | Zbl

[KKMS] Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B., Toroidal embeddings, Lecture Notes in Math., 339, (1973). | Zbl

[KR] Kewenig, A., Rieband, Th.: Geisterklassen im Bild der Borelabbildung für symplektishe und orthogonale Gruppen, Diplomarbeit, Uni. Bonn, 1997.

[K] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math., 74, 329–287 (1961). | Zbl

[Le] Lepowsky, J., A generalization of the Bernstein-Gelfand-Gelfand resolution, J. Algebra, 49, 496–511 (1977). | Zbl

[L] Looijenga, E., $L_{2}$ -cohomology of locally symmetric varieties, Compositio Math., 67, 3–20 (1988). | EuDML | Zbl | Numdam

[M1] Milne, J.S., Automorphic vector bundles on connected Shimura varieties, Invent. Math., 92, 91–128 (1988). | EuDML | Zbl

[M2] Milne, J.S., Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Clozel, L., Milne, J. (eds.), Automorphic Forms, Shimura Varieties, and $L$ -functions, v. 1, 283–414 (1990). | Zbl

[M3] Milne, J.S., Descent for Shimura varieties. Michigan Math. J., 46, 203-208 (1999). | Zbl

[Mo] Moonen, B., Models of Shimura varieties in mixed characteristics, in A. J. Scholl and R. L. Taylor, eds., Galois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Note Series, 254, 267-350 (1998). | Zbl

[Mu] Mumford, D., Hirzebruch’s proportionality theorem in the non-compact case, Inventiones Math., 42, 239–272 (1977). | EuDML | Zbl

[P] Pink, R., Arithmetical compactifications of mixed Shimura varieties. Bonn Math. Schrift., Nr. 209 (1990).

[RC] Rocha-Caridi, A., Splitting criteria for $𝔤$ -modules induced from a parabolic and the Bernstein-Gelfand-Gelfand resolution of a finite dimensional, irreducible $𝔤$ -module, Trans. Am. Math. Soc., 262, 335–366 (1980). | Zbl

[Sa1] Saito, MoSaito, Mo., Modules de Hodge polarisables. Publ. RIMS Kyoto Univ., 24 (1988), 849–995.

[Sa2] Saito, MoSaito, Mo.: Mixed Hodge modules and admissible variations. C.R.A.S. Paris, 309, 351–356 (1989).

[Sa3] Saito, MoSaito, Mo., Mixed Hodge modules, Publ. RIMS Kyoto Univ. 26, 221–333 (1990).

[SS] Saper, L., Stern M., $L_{2}$ -cohomology of arithmetic varieties. Ann. of Math. 132 (1990), 1–69. | Zbl

[SZ] Saper, L., Zucker, S., An introduction to $L^{2}$ -cohomology. Several Complex Variables and Complex Geometry, Santa Cruz, 1989 Proc. of Symposia in Pure Math. 52 (1991), Part 2, 519–534. | Zbl

[Sch] Schwermer, J., Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen, Lecture Notes in Math., 988, (1983). | Zbl

[SGA 3] Demazure, M., Grothendieck, A., Schémas en groupes, Lecture Notes in Math., 151, 152, 153 (1970). | Zbl

[StZ] Steenbrink, J., Zucker, S., Variation of mixed Hodge structure, I. Inventiones Math., 80, 489–542, (1985). | EuDML | Zbl

[W1] Wildeshaus, J., Realizations of polylogarithms, Lecture Notes in Math., 1650, (1997). | Zbl

[W2] Wildeshaus, J., Mixed sheaves on Shimura varieties and their higher direct images in toroidal compactifications, J. Alg. Geom. 9 (2000), 323–353. | Zbl

[WK] Wolf, J. Kóranyi, A., Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math., 87, 899–939 (1965).

[Z1] Zucker, S., Locally homogeneous variations of Hodge structure. L’Enseignement Math. 27, 243–275, (1981). | Zbl

[Z2] Zucker, S., $L_{2}$ cohomology of warped products and arithmetic groups. Inventiones Math. 70, 169–218, (1982). | EuDML | Zbl

[Z3] Zucker, S., Degeneration of mixed Hodge structures. Algebraic Geometry, Proc. of Symposia in Pure Math., 46, 283–293, (1987).

[Z4] Zucker, S., The Hodge structures on the intersection homology of varieties with isolated singularities. Duke Math. J. 55, (1987), 603–616. | Zbl

[Z5] Zucker, S., On the boundary cohomology of locally symmetric varieties. Vietnam J. Math. 25, 279–318, (1997).

[Z6] Zucker, S., $L_{2}$ cohomology and intersection homology of locally symmetric varieties, II. Compositio Math. 59, (1986), 339–398. | EuDML | Zbl | Numdam

Cité par Sources :