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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.

##### MSC:
 32F45 Invariant metrics and pseudodistances in several complex variables
##### Keywords:
Bergman metric; potential; homogeneous domains; Siegel domains
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##### References:
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