Harris, Michael; Li, Jian-Shu A Lefschetz property for subvarieties of Shimura varieties. (English) Zbl 0954.14016 J. Algebr. Geom. 7, No. 1, 77-122 (1998). Summary: Let \(H\subset G\) be reductive groups over \(\mathbb{Q}\), \(\Gamma \subset G(\mathbb{R})\) an arithmetic subgroup. Let \(\pi\), respectively \(\pi'\), be irreducible unitary representations of \(G(\mathbb{R})\), respectively \(H (\mathbb{R})\), with \(\pi\) cohomological and \(\pi'\) in the discrete series. Suppose \(\pi\) is isomorphic to a subspace \(\sigma\) of the space of automorphic forms on \(\Gamma \setminus G(R)\) and \(\pi'\) is isomorphic to a direct factor of the restriction of \(\pi\) to \(H(\mathbb{R})\). Assuming the symmetric spaces \(X_G\) \(X_H\) associated to \(G\) and \(H\), respectively, are of Hermitian type, we apply recent results of M. Burger and P. Sarnak [Invent. Math. 106, No. 1, 1-11 (1991; Zbl 0774.11021)] to show in some cases that the natural restriction from the cohomology of \(\Gamma\setminus X_G\) to the product of cohomologies of spaces of the form \(\Gamma_H\setminus X_H\) is injective. To construct pairs \(\pi,\pi'\) with the necessary properties we combine functional analytic techniques with a generalization of M. Flensted-Jensen’s [Ann. Math. (2) 111, 253-311 (1980; Zbl 0462.22006)] formula for discrete series matrix coefficients. To apply the Burger-Sarnak theorem we appeal to the theory of base change of automorphic forms and to a theorem of W. Luo, J. Rudnick and P. Sarnak [Geom. Funct. Anal. 5, No. 2, 387-401 (1995; Zbl 0844.11038)] on the contribution of the complementary series of \(GL(n, \mathbb{C})\) to the automorphic spectrum.We specifically treat groups \(G\) of type \(U(n,1)\) or \(SO(n,2)\). Assuming \(\Gamma\) cocompact, we obtain unconditional results on the cohomology in degree 2, generalizing results of T. Oda [J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 481-486 (1981; Zbl 0528.14022)] and V. K. Murty and D. Ramakrishnan [in: The zeta function of Picard modular surfaces, CRM Workshop, Montreal 1988, 445-464 (1992; Zbl 0828.14013)] for cohomology in degree 1. In the \(U(n,1)\) case we obtain results on cohomology in degrees \(<n\) assuming a standard conjecture on base change. The techniques of this article also provide analytic criteria for rationality of coherent cohomology classes. Cited in 2 ReviewsCited in 18 Documents MSC: 14G35 Modular and Shimura varieties 14M17 Homogeneous spaces and generalizations 14K12 Subvarieties of abelian varieties 14F20 Étale and other Grothendieck topologies and (co)homologies Keywords:subvarieties of Shimura varieties; space of automorphic forms; symmetric spaces; cohomology; base change Citations:Zbl 0774.11021; Zbl 0462.22006; Zbl 0844.11038; Zbl 0528.14022; Zbl 0828.14013 PDFBibTeX XMLCite \textit{M. Harris} and \textit{J.-S. Li}, J. Algebr. Geom. 7, No. 1, 77--122 (1998; Zbl 0954.14016)