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This paper considers a class of homogeneous Kahler metrics which include the Bergman metrics on homogeneous bounded domains. We obtain various necessary conditions for (a) nonpositive holomorphic sectional curvature, (b) nonpositive sectional curvature, and (c) covariant constant curvature (symmetric metric). In particular, we give examples showing that there exist homogeneous bounded domains which in the Bergman metric have some positive holomorphic sectional curvature.

Holomorphic sectional curvatures of bounded homogeneous domains and related questions