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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.

##### 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