The Gromov norm of the Kähler class of symmetric domains (Q1077747)

The purpose of this paper is prove that if X is a compact complex manifold whose universal cover is a bounded symmetric domain of rank p and \(\omega\) is the Kähler form of the metric on X induced by the Bergmann metric of the domain (normalized so that the minimum holomorphic sectional curvature is -1), then the sup norm of the cohomology class of \(\omega\) in the sense of Gromov is \(p\pi\). This is proved as follows: An upper bound is found by evaluating explicit integrals involving the kernel function. A lower bound is found by evaluating on arithmetic examples.