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Boundary cohomology of Shimura varieties

II. Hodge theory at the boundary

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References

  • [AMRT] Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth Compactification of Locally Symmetric Varieties. Brookline, MA: Math. Sci. Press 1975

    Google Scholar 

  • [BB] Baily, W.L. Jr., Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math.84, 442–528 (1966)

    Google Scholar 

  • [B] Borel, A.: Cohomology and spectrum of an arithmetic group. In: Operator Algebras and Group Representations. Proc. Neptun, Romania, 1980, Pitman, pp. 28–45. Boston London Melborne: Pitman (1984)

    Google Scholar 

  • [BS] Borel, A., Serre, J.-P.: Corners and arithmetic groups. Comment. Math. Helv.48, 436–491 (1973)

    Google Scholar 

  • [CK] Cattani, E., Kaplan, A.: Polarized mixed Hodge structure and the monodromy of a variation of Hodge structure. Invent. Math.67, 101–115 (1982)

    Google Scholar 

  • [D1] Deligne, P.: Equations Differentielles à Points Singuliers Réguliers. Berlin Heidelberg New York: Springer Lect. Notes Math., vol. 163, (1970)

  • [D2] Deligne, P.: Théorie de Hodge. II. Publ. Math. IHES40, 5–57 (1972)

    Google Scholar 

  • [D3] Deligne, P.: Théorie de Hodge. III. Publ. Math. IHES44, 5–77 (1974)

    Google Scholar 

  • [D4] Deligne, P.: Variétés de Shimura: Interprétation modulaire, et techniques de construction de modèles canoniques. In: Automorphic Forms, Representations, andL-functions. Proc. Symp. Pure Math.33, 247–290 (1979)

  • [D5] Deligne, P.: La conjecture de Weil. II. Publ. Math. IHES52, 137–252 (1980)

    Google Scholar 

  • [D6] Deligne, P.: Theorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. IHES35, 107–126 (1969)

    Google Scholar 

  • [E] El Zein, F.: Complexe de Hodge mixte filtré. C.R. Acad. Sci., Paris295, 669–672 (1982)

    Google Scholar 

  • [Hd1] Harder, G.: On the cohomology of discrete arithmetically defined groups. In: Proc. of the Int. Colloq. on Discrete Subgroups of Lie Groups and Applications to Moduli, Oxford University Press, pp. 129–160, 1975

  • [Hd2] Harder, G.: Some results on the Eisenstein cohomology of arithmetic subgroups ofGL n . In: Labesse, J.-P., Schwermer, J. (eds.), Cohomology of Arithmetic Groups and Automorphic Forms. Berlin Heidelberg New York: Springer Lect. Notes Math. vol.1447, 85–153 (1990)

  • [Hd3] Harder, G.: Eisenstein cohomology of arithmetic groups and its applications to number theory. In: Proc. ICM (Kyoto, 1990), 779–790.

  • [H1] Harris, M.: Functorial properties of toroidal compactifications of locally symmetric varieties. Proc. London Math. Soc.59, 1–22 (1989)

    Google Scholar 

  • [H2] Harris, M.: Arithmetic vector bundles and automorphic forms on Shimura varieties, II. Compos. Math.60, 323–378 (1986)

    Google Scholar 

  • [H3] Harris, M.: Hodge-de Rham structures and periods of automorphic forms. In: Motives, Seattle, 1991, Proc. Symp. Pure Math. (to appear)

  • [HZ] Harris, M., Zucker, S.: Boundary cohomology of Shimura varieties, I: Coherent cohomology on toroidal compactifications. Ann. Ec. Norm. Super. (to appear)

  • [He] Helgason, S.: Differential Geometry. Lie Groups, and Symmetric Spaces. London New York: Academic Press, 1978

    Google Scholar 

  • [Ka] Kashiwara, M.: A study of variation of mixed Hodge structure. Publ. RIMS, Kyoto Univ.22, 991–1024 (1986)

    Google Scholar 

  • [Ko] Kostant, B.: Lie algebra cohomology and the generalized Borel-Weil theorem. Ann. Math.74, 329–387 (1961)

    Google Scholar 

  • [LR] Looijenga, E., Rapoport, M.: Weights in the local cohomology of a Baily-Borel compactification. In: Complex Geometry and Lie Theory. Proc. Symp. Pure Math.53, 223–260 (1991)

  • [Mi] Milne, J.: Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Clozel, L., Milne, J. (eds.) Automorphic Forms, Shimura Varieties, andL-functions, vol. 1, 283–414 (1990)

  • [Mo] Moore, C.: Compactifications of symmetric spaces II: the Cartan domains. Am. J. Math.86, 358–378 (1964)

    Google Scholar 

  • [Mu] Mumford, D.: Hirzebruch's proportionality theorem in the non-compact case. Invent. Math.42, 239–272 (1977)

    Google Scholar 

  • [Na] Navarro Aznar, V.: Sur la théorie de Hodge-Deligne. Invent. Math.90, 11–76 (1987)

    Google Scholar 

  • [No] Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math.59, 531–538 (1954)

    Google Scholar 

  • [O] Oda, T.: Periods of Hilbert modular surfaces. Prog. Math. vol. 19, Birkhäuser, Boston (1982)

    Google Scholar 

  • [OS] Oda, T., Schwermer, J.: Mixed Hodge structures and automorphic forms for Siegel modular varieties of degree two. Math. Ann.286, 481–509 (1990)

    Google Scholar 

  • [P1] Pink, R.: Arithmetical compactifications of mixed shimura varieties. Dissertation, Universität Bonn, 1989 (Bonn Math. Schrift Nr. 209 (1990))

  • [P2] Pink, R.: On ℓ-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification. Math. Ann.292, 197–240 (1992)

    Google Scholar 

  • [R] Rapoport, M.: Letter to M. Goresky and R. MacPherson, August14, 1991

  • [S1] Saito, Mo: Introduction to mixed Hodge modules. Théorie de Hodge: Luminy, Juin 1987, Astérisque179–180, 145–162 (1989)

    Google Scholar 

  • [S2] Saito, Mo: Mixed Hodge modules and admissible variations. C.R.A.S. Paris309, 351–356 (1989)

    Google Scholar 

  • [S3] Saito, Mo.: Mixed Hodge modules. Pub. RIMS, Kyoto University26, 221–333 (1990)

    Google Scholar 

  • [SZ] Saito, Mo., Zucker, S.: The kernel spectral sequence of vanishing cycles. Duke Math. J.61, 329–339 (1990)

    Google Scholar 

  • [Sa] Satake, I.: On compactifications of the quotient spaces for arithmetically defined discontinuous groups. Ann. Math.72, 555–580 (1960)

    Google Scholar 

  • [Sc] Schmid, W.: Variation of Hodge structure: The singularities of the period mapping. Invent. Math.22, 211–319 (1973)

    Google Scholar 

  • [Sw] Schwermer, J.: Kohomologie arithmetisch definierter Gruppen und Eisensteinreihen. Berlin Heidelberg New York: Springer (Lect. Notes Math. vol. 988, 1983)

    Google Scholar 

  • [Se] Serre, J.-P.: Geométrie algébrique et géométrie analytique. Ann. Inst. Fourier6, 1–42 (1956)

    Google Scholar 

  • [StZ] Steenbrink, J., Zucker, S.: Variation of mixed Hodge structure. I. Invent. Math.80, 489–542 (1985)

    Google Scholar 

  • [Sn] Stern, M.: Harmonic representatives of de Rham cohomology on locally symmetric spaces, 1992

  • [Z1] Zucker, S.: Hodge theory with degenerating coefficients:L 2 cohomology in the Poincaré metric. Ann. Math.109, 415–476 (1979)

    Google Scholar 

  • [Z2] Zucker, S.: Locally homogeneous variations of Hodge structure. L'Enseignement Math.27, 243–276 (1981)

    Google Scholar 

  • [Z3] Zucker, S.:L 2 cohomology of warped products and arithmetic groups. Invent. Math.70, 169–218 (1982)

    Google Scholar 

  • [Z4] Zucker, S.: Satake compactifications. Comment. Math. Helv.58, 312–343 (1983)

    Google Scholar 

  • [Z5] Zucker, S.:L 2 cohomology and intersection homology of locally symmetric varieties, II. Compos. Math.59, 339–398 (1986)

    Google Scholar 

  • [Z6] Zucker, S.:L p-cohomology and Satake compactifications. In: J. Noguchi, T. Ohsawa (eds.), Prospects in Complex Geometry: Proceedings, Katata/Kyoto 1989. Berlin Heidelberg New York: Springer (Lect. Notes Math. vol. 1468, pp. 317–339) 1991

    Google Scholar 

  • [Z7] Zucker, S.:L 2-cohomology of Shimura varieties. In: Clozel, L., Milne, J. (eds.) Automorphic Forms, Shimura Varieties, andL-functions, vol. 2, 377–391 (1990)

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Dedicated to Armand Borel

Oblatum 15-III-1993

Supported in part by the National Science Foundation, through Grants DMS-8901101 and DMS-9203142

Supported in part by the National Science Foundation, through Grant DMS-9102233

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Harris, M., Zucker, S. Boundary cohomology of Shimura varieties. Invent Math 116, 243–308 (1994). https://doi.org/10.1007/BF01231562

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