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A Lefschetz property for subvarieties of Shimura varieties. (English) Zbl 0954.14016
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.

14G35 Modular and Shimura varieties
14M17 Homogeneous spaces and generalizations
14K12 Subvarieties of abelian varieties
14F20 √Čtale and other Grothendieck topologies and (co)homologies