  The main result of this note essentially is that if the base and fibers of a
compact fibration carry Hermitian metrics of positive holomorphic sectional
curvature, then so does the total space of the fibration. The proof is based on
the use of a warped product metric as in the work by Cheung in case of negative
holomorphic sectional curvature, but differs in certain key aspects, e.g., in
that it does not use the subadditivity property for holomorphic sectional
curvature due to Grauert-Reckziegel and Wu.
