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Lefschetz properties for arithmetic real and complex hyperbolic manifolds. (English) Zbl 1036.11021
Let \(M\) be a projective variety of complex dimension \(n\). Then the complex cohomology and homology groups \(H^* (M)\) and \(H_* (M)\) can be understood in every, but the middle, dimension by looking at a smaller projective space. Indeed, if \(H\) is a generic hyperplane in the ambient projective space, the Lefschetz theorem shows the existence of the embeddings \(H_i (M\cap H) \hookrightarrow H_i (M)\) and \(H^j (M) \hookrightarrow H^j (M\cap H)\) for \(i \geq n\) and \(j \leq n-1\). Let \(G\) be a semisimple algebraic group over \(\mathbb Q\), and let \(X= G(\mathbb R) /K_\infty\) with \(K_\infty\) being a maximal compact subgroup of \(G(\mathbb R)\).
In this paper the author provides an analogous, but mostly conjectural, result for the quotient \(M^d = \Gamma \backslash X\) of \(X\) by an arithmetic subgroup \(\Gamma\) of \(G(\mathbb Q)\). He states a conjecture, discusses its evidence, and describes some results implied by the conjecture.

MSC:
11F75 Cohomology of arithmetic groups
14G35 Modular and Shimura varieties
14F25 Classical real and complex (co)homology in algebraic geometry
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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