Abstract
We obtain a necessary condition for a cohomology class on a compact locally symmetric space S(Γ)=Γ∖X (a quotient of a symmetric space X of the non-compact type by a cocompact arithmetic subgroup Γ of isometries of X) to restrict non-trivially to a compact locally symmetric subspace S H(Γ)=Δ∖Y of Γ∖X. The restriction is in a 'virtual' sense, i.e. it is the restriction of possibly a translate of the cohomology class under a Hecke correspondence. As a consequence we deduce that when X and Y are the unit balls in ℂn and ℂm, then low degree cohomology classes on the variety S(Γ) restrict non-trivially to the subvariety S H (Γ); this proves a conjecture of M. Harris and J-S. Li. We also deduce the non-vanishing of cup-products of cohomology classes for the variety S(Γ).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arthur, J.: Unipotent automorphic representations, conjectures, Astèrisque 171–172(1989), 13–71 Soc Math. France
Baily, W. and Borel, A.: Compacti¢cations of arithmetic quotients of bounded sym-metric domains, Ann. Math. 84 (1966), 442–528.
Borel, A. and Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Ann. Math. Stud., Princeton Univ. Press, 1980.
Clozel, L.: Produits dans la cohomologie, J. Reine Angew. Math., 430 (1992), 69–83
Clozel, L.: Produits dans la cohomologie II, J. Reine Angew.Math. 444 (1993), 1–15.
Clozel, L.: Motifs et formes automorphes, applications de functorialité, In: Auto-morphic Forms, Shimura Varieties and L-functions, Perspect. Math. Vol (I), Academic Press, New York.
Clozel, L. and Venkataramana, T. N.: Restriction of holomorphic cohomology of a Shimura variety to a smaller Shimura variety, Duke J. Math. 95(1) (1998), 51–106.
Deligne, P.: Variétés de Shimura, interprétation modulaire, Proc. Sympos. PureMath. 33(2) (1979), 247–290.
Fulton, W. and Harris, J.: Representation Theory, Springer, New York, 1991.
Griffiths, P. and Harris, J.: Principles of Algebraic Geometry, Pure Appl. Math., Wiley, New York, 1978.
Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.
Harris, M. and Li, J-S.: A Lefschetz property for subvarieties of Shimura varieties, J. Algebraic Geom., 1998.
Kostant, B.: Lie Algebra cohomology and generalised Schubert cells, Ann. Math. 77(1) (1963), 72–144.
Kudla, S. S.: Algebraic cycles of Shimura varieties of orthogonal type, Duke Math. J. 86(1) (1997), 39–78.
Kumaresan, S.: The canonical k-types of irreducible (g,K)-modules with non-zero cohomology, Invent Math. 59 (1980), 1–11.
Mostow, G. D.: Self adjoint groups, Ann. Math. 62 (1955), 44–55.
Kumar Murthy, V. and Ramakrishnan, D.: The Albanese of unitary Shimura varieties, In: Langlands and Ramakrishnan (eds), Zeta Functions of Picard Modular Surfaces, Publ. CRM, Montreal, 1992.
Oda, T.: A note on the Albanese of certain unitary Shimura varieties J. Fac. Sci. Univ. Tokyo, Section 1A, Math. 28 (1981), 481–486.
Parthasarathy, R.: Criteria for the unitarisability of certain highest weight modules, Proc. Ind. Acad. Sci. 89 (1980), 1–24.
Parthasarathy, R.: Holomorphic forms on г\G/K and Chern classes, Topology 21 (1982), 152–175.
Platonov, V. P.: Strong approximation in algebraic groups and the Kneser Tits con-jecture, Dokl Akad. Nauk BSSR 13(7) (1969), 585–587.
Prasad, G.: Strong approximation for demisimple groups over function fields, Ann. Math. 105(3) (1977), 553–572.
Raghunathan, M. S.: Discrete Subgroups of Lie Groups, Ergeb. Math., Springer, Berlin, 1972.
Serre, J.-P.: Geomètrie algébrique et geométrie analitique, Ann. Inst. Fourier 6 (1956), 1–42.
Shimura, G.: Unitary groups and theta functions, In: Proc. Int. Sympos. Algebra and N. Theory, (Kyoto, 1976), pp. 195–200.
Shimura, G.: Arithmetic of automorphic forms with respect to a unitary group, Ann Math. 2 107(3) (1978), 563–605.
Shimura, G.: Automorphic forms and periods of Abelian varieties, J. Math. Soc. Japan 31 (1979), 561–592
Vogan, D. and Zuckerman, G.: Unitary representations and cohomology,Compositio Math. 53 (1984), 51–90.
Weissauer, R.: Differential formen Zu Unterguppen der Siegelsche Modulgruppe Zweiten Grades, J. reine angew. Math. 391 (1988), 100–156.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Venkataramana, T.N. Cohomology of Compact Locally Symmetric Spaces. Compositio Mathematica 125, 221–253 (2001). https://doi.org/10.1023/A:1002600432171
Issue Date:
DOI: https://doi.org/10.1023/A:1002600432171