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AUTOMORPHIC LEFSCHETZ PROPERTIES FOR NONCOMPACT ARITHMETIC MANIFOLDS
Part of:
Arithmetic problems. Diophantine geometry
Arithmetic algebraic geometry
Discontinuous groups and automorphic forms
Published online by Cambridge University Press: 18 October 2021
Abstract
We prove the injectivity of Oda-type restriction maps for the cohomology of noncompact congruence quotients of symmetric spaces. This includes results for restriction between (1) congruence real hyperbolic manifolds, (2) congruence complex hyperbolic manifolds, and (3) orthogonal Shimura varieties. These results generalize results for compact congruence quotients by Bergeron and Clozel [Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math. 192 (2013), 505–532] and Venkataramana [Cohomology of compact locally symmetric spaces, Compos. Math. 125 (2001), 221–253]. The proofs combine techniques of mixed Hodge theory and methods involving automorphic forms.
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 4 , July 2023 , pp. 1655 - 1702
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
Arthur, J., Unipotent automorphic representations: Conjectures, Astérisque 171-172 (1989), 13–71.Google Scholar
Arthur, J., The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, AMS Colloquium Publications, 61 (American Mathematical Society, Providence, RI, 2013).Google Scholar
Ash, A., Mumford, D., Rapoport, M. and Tai, Y.-S., Smooth Compactification of Locally Symmetric Varieties, 2nd ed. (Cambridge University Press, Cambridge, UK, 2010).10.1017/CBO9780511674693CrossRefGoogle Scholar
Atiyah, M. F. and Hirzebruch, F., Vector bundles and homogeneous spaces, in Differential Geometry, Proceedings of Symposia in Pure Mathematics, 3, pp. 7–38 (American Mathematical Society, Providence, RI, 1961).CrossRefGoogle Scholar
Baily, W. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528.CrossRefGoogle Scholar
Beilinson, A., Bernstein, J. and Deligne, P., Faisceaux Pervers, Astérisque 100 (1982).Google Scholar
Bergeron, N., Lefschetz properties for arithmetic real and complex hyperbolic manifolds, Int. Math. Res. Not. IMRN 20 (2003) 1089–1122.10.1155/S1073792803212253CrossRefGoogle Scholar
Bergeron, N., Propriétés de Lefschetz automorphes pour les groupes unitaires et orthogonaux, Mém. Soc. Math. Fr. (N.S.) 106 (2006).Google Scholar
Bergeron, N., Restriction de la cohomologie d’une variété de Shimura à ses sous-variétés, Transform. Groups 14 (2009), 41–86.10.1007/s00031-008-9047-4CrossRefGoogle Scholar
Bergeron, N., Hodge theory and cycle theory of locally symmetric spaces, in Proceedings of the International Congress of Mathematicians 2018, pp. 831–857 (World Scientific, Singapore, 2019).CrossRefGoogle Scholar
Bergeron, N. and Clozel, L., Spectre automorphe des variétés hyperboliques et applications topologiques Astérisque 303 (2005).Google Scholar
Bergeron, N. and Clozel, L., Quelques conséquences des travaux d’Arthur pour le spectre et la topologie des variétés hyperboliques, Invent. Math. 192 (2013), 505–532.Google Scholar
Bergeron, N. and Clozel, L., Sur le spectre et la topologie des variétés hyperboliques de congruence: Les cas complexe et quaternionien, Math. Ann. 368 (2017), 1333–1358.10.1007/s00208-016-1492-0CrossRefGoogle Scholar
Bergeron, N. and Clozel, L., Sur la cohomologie des variétés hyperboliques de dimension
$7$
trialitaires, Israel J. Math. 222 (2017), 333–400.CrossRefGoogle Scholar
$7$
trialitaires, Israel J. Math. 222 (2017), 333–400.CrossRefGoogle ScholarBorcherds, R. E., Automorphic forms on
${O}_{s+2,2}\left(\mathbf{R}\right)$
and infinite products, Invent. Math. 120 (1995), 161–213.10.1007/BF01241126CrossRefGoogle Scholar
${O}_{s+2,2}\left(\mathbf{R}\right)$
and infinite products, Invent. Math. 120 (1995), 161–213.10.1007/BF01241126CrossRefGoogle ScholarBorel, A. and Casselman, W.,
${L}^2$
cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), 625–647.CrossRefGoogle Scholar
${L}^2$
cohomology of locally symmetric manifolds of finite volume, Duke Math. J. 50 (1983), 625–647.CrossRefGoogle ScholarBorel, A. and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, 2nd ed., Mathematical Surveys and Monographs, 67 (American Mathematical Society, Providence, RI, (1999).Google Scholar
Burger, M. and Sarnak, P., Ramanujan duals II, Invent. Math. 106 (1991), 1–11.10.1007/BF01243900CrossRefGoogle Scholar
Burgos, J. I. and Wildeshaus, J., Hodge modules on Shimura varieties and their higher direct images in the Baily-Borel compactification, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 363–413.10.1016/j.ansens.2004.01.002CrossRefGoogle Scholar
Deligne, P., Théorie de Hodge III, Publ. Math. Inst. Hautes Études Sci. 44 (1974), 5–77.10.1007/BF02685881CrossRefGoogle Scholar
Franke, J., Harmonic analysis in weighted
${L}_2$
spaces, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 181–279.CrossRefGoogle Scholar
${L}_2$
spaces, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 181–279.CrossRefGoogle ScholarGoodman, R. and Wallach, N., Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, 255 (Springer, New York, NY, 2009).10.1007/978-0-387-79852-3CrossRefGoogle Scholar
Goresky, M., Harder, G., and MacPherson, R., Weighted cohomology, Invent. Math. 116 (1994), 139–213.CrossRefGoogle Scholar
Goresky, M. and MacPherson, R., Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 14 (Springer-Verlag, New York, NY, 1988).CrossRefGoogle Scholar
Goresky, M. and Pardon, W., Chern classes of automorphic vector bundles, Invent. Math. 147 (2002), 561–612.CrossRefGoogle Scholar
Harder, G., On the cohomology of discrete arithmetically defined groups, in Discrete Subgroups of Lie Groups (Bombay, 1973), pp. 129–160 (Oxford University Press, Bombay, 1975).Google Scholar
Harris, M., Functorial properties of toroidal compactifications of locally symmetric varieties, Proc. Lond. Math. Soc. (3) 59 (1989), 1–22.10.1112/plms/s3-59.1.1CrossRefGoogle Scholar
Harris, M. and Li, J.-S., A Lefschetz property for subvarieties of Shimura varieties, J. Algebraic Geom. 7 (1998), 77–122.Google Scholar
Kostant, B., Lie algebra cohomology and the Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329–387.CrossRefGoogle Scholar
Looijenga, E.,
${L}^2$
cohomology of locally symmetric varieties, Compos. Math. 23 (1988), 1–20.Google Scholar
${L}^2$
cohomology of locally symmetric varieties, Compos. Math. 23 (1988), 1–20.Google ScholarLooijenga, E. and Rapoport, M., Weights in the local cohomology of a Baily-Borel compactification, in Complex Geometry and Lie Theory, Proceedings of Symposia in Pure Mathematics, 53, pp. 223–260 (American Mathematical Society, Providence, RI, 1991).10.1090/pspum/053/1141203CrossRefGoogle Scholar
Mumford, D., Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math. 42 (1977), 239–272.CrossRefGoogle Scholar
Nair, A., Weighted cohomology of arithmetic groups, Ann. of Math. (2) 150 (1999), 1–31.10.2307/121096CrossRefGoogle Scholar
Nair, A., ‘Chern classes of automorphic vector bundles and the reductive Borel-Serre compactification’, Preprint, 2014, http://www.math.tifr.res.in/~arvind.Google Scholar
Nair, A., Lefschetz properties for noncompact arithmetic ball quotients, J. Reine Angew. Math. 730 (2017), 163–198.CrossRefGoogle Scholar
Nair, A., Lefschetz properties for noncompact arithmetic ball quotients II, Manuscripta Math. 152 (2017), 443–457.10.1007/s00229-016-0868-5CrossRefGoogle Scholar
Nair, A., Motivic and automorphic aspects of the reductive Borel-Serre compactification, in
Hodge Theory and L
2 Analysis (Volume in Honour of S. Zucker)
, Advanced Lectures in Mathematics, 39, pp. 473–485 (International Press, Somerville, MA, 2017).Google Scholar
Oda, T., A note on the Albanese variety of an arithmetic quotient of the complex hyperball, J. Fac. Sci. Univ. Tokyo 28 (1981), 481–486.Google Scholar
Pink, R., On
$\ell$
-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292 (1992), 197–240.CrossRefGoogle Scholar
$\ell$
-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292 (1992), 197–240.CrossRefGoogle ScholarPittie, H. V., Homogeneous vector bundles on homogeneous spaces, Topology 11 (1972), 199–203.10.1016/0040-9383(72)90007-9CrossRefGoogle Scholar
Rohlfs, J. and Speh, B., Representations with cohomology in the discrete spectrum of
$\mathrm{SO}\left(n,1\right)\left(\mathbb{Z}\right)$
and Lefschetz numbers, Ann. Sci. Éc. Norm. Supér. (4) 20 (1987), 89–136.CrossRefGoogle Scholar
$\mathrm{SO}\left(n,1\right)\left(\mathbb{Z}\right)$
and Lefschetz numbers, Ann. Sci. Éc. Norm. Supér. (4) 20 (1987), 89–136.CrossRefGoogle ScholarSaito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), 221–333.CrossRefGoogle Scholar
Saper, L. and Stern, M.,
${L}^2$
cohomology of arithmetic varieties, Ann. of Math. (2) 139 (1990), 1–69.CrossRefGoogle Scholar
${L}^2$
cohomology of arithmetic varieties, Ann. of Math. (2) 139 (1990), 1–69.CrossRefGoogle ScholarSatake, I., Algebraic Structures of Symmetric Domains, Iwanami Shoten, Tokyo; (Princeton University Press, Princeton, NJ, 1980).Google Scholar
Schwermer, J., Kohomologie arithmetisch definierte Gruppen und Eisensteinreihen, Lecture Notes in Mathematics, 988 (Springer-Verlag, New York, NY, 1983).CrossRefGoogle Scholar
Tits, J., Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics, 9, pp. 33–62 (American Mathematical Society, Providence, RI, 1966).CrossRefGoogle Scholar
Venkataramana, T. N., Cohomology of compact locally symmetric spaces, Compos. Math. 125 (2001), 221–253.CrossRefGoogle Scholar
Venkataramana, T. N., Cohomology of arithmetic groups and representations, in Proceedings of the International Congress of Mathematicians 2010
, Vol. III, pp. 1366–1375 (Hindustan Book Agency, New Delhi, 2010).Google Scholar
Waldspurger, J.-L., Cohomology des espaces des formes automorphes (d’aprés J. Franke), Séminaire Bourbaki 1995/1996, Exp. No. 809, Astérisque 241 (1997), 139–156.Google Scholar
Wallach, N., On the constant term of a square-integrable automorphic form, in Operator Algebras and Group Representations
, Vol. II (Neptun, 1980), Monographs and Studies in Mathematics, 18, pp. 227–237 (Pitman, Boston, MA, 1984).Google Scholar
Weissauer, R., Differentialformen zu Untergruppen der Siegelschen Modulgruppe zweiten Grades, J. Reine Angew. Math. 391 (1988), 100–156.Google Scholar
Zucker, S., The Hodge structures on the intersection homology of varieties with isolated singularities, Duke Math. J. 55 (1987), 603–616.CrossRefGoogle Scholar