Cohomology of compact locally symmetric spaces (Q2711524)

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 \textit{M. Harris} and \textit{J.-S. Li} [J. Algebr. Geom. 7, 77-122 (1998; Zbl 0954.14016)].