zbMATH — the first resource for mathematics

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.

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.)
Full Text: DOI