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Cohomology of compact locally symmetric spaces. (English) Zbl 0983.11027
Let $$\Gamma\setminus X$$ be a quotient of a symmetric space by a congruence arithmetic group, and let $$\Delta\setminus Y$$ be a subspace of the same kind. The main result of the paper is that a cohomology class $$\alpha$$ of $$\Gamma\setminus X$$ restricts to zero on $$\Delta\setminus Y$$ only if the cup product $$\alpha\cup [\widehat{Y}]$$ vanishes where $$[\widehat{Y}]$$ is the cycle class of the compact dual $$\widehat{Y}$$ of the symmetric space $$Y$$. As an application, it is deduced that if $$X$$ and $$Y$$ are the unit balls in $$\mathbb{C}^n$$ and $$\mathbb{C}^m$$, then the low degree classes of $$\Gamma\setminus X$$ restrict nontrivially to the subvariety $$\Delta\setminus Y$$. This proves a conjecture of M. Harris and J.-S. Li [J. Algebr. Geom. 7, 77-122 (1998; Zbl 0954.14016)].

##### MSC:
 11F75 Cohomology of arithmetic groups 22E40 Discrete subgroups of Lie groups
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