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A note on the Bergman metric of bounded homogeneous domains. (English) Zbl 1127.32008
The authors study the Bergman metric \(ds_D^2\) on a bounded homogeneous domain \(D\) in \(\mathbb C^n\). They prove first, that there exists a real-analytic potential \(\varphi\) for \(ds_D^2\), whose gradient has constant length \(C\), when measured with respect to the Bergman metric, and, further, that this constant \(C\) is independent of the choice of \(\varphi\). As an application, they conclude that the \(L^2\) \(\overline{\partial} \) cohomology of \(D\) (with respect to the Bergman metric) at the bidegree \((p,q)\) vanishes for \(p+q\neq n\) and has infinite dimension for \(p+q=n\) and is Hausdorff.

32F45 Invariant metrics and pseudodistances in several complex variables
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