  We study pseudonorms on pluricanonical bundles over Stein manifolds. We prove
that the pseudonorms determine holomorphic structures of Stein manifolds under
certain assumptions. This theorem is based on and a generalization of the
result obtained by Deng, Wang, Zhang and Zhou \cite{DWZZ} for bounded domains
in $\mathbb{C}^n$. We also investigate Stein morphisms and the pseudonorms on
direct images of pluricanonical bundles. Our main goal in this paper is to show
that the pseudonorms also determine holomorphic structures of Stein morphisms.
One important technique is an $L^{2/m}$-variant of the Ohsawa-Takegoshi
extension theorem.

Complex Variables;Algebraic Geometry