Pseudonorms on direct images of pluricanonical bundles

TL;DR

This paper investigates how pseudonorms on pluricanonical bundles can determine the holomorphic structures of Stein manifolds and morphisms, extending previous results and employing an $L^{2/m}$-variant of the Ohsawa-Takegoshi extension theorem.

Contribution

It generalizes existing results by showing pseudonorms determine holomorphic structures for Stein morphisms and introduces an $L^{2/m}$-variant of the extension theorem.

Findings

Pseudonorms determine Stein manifold structures.

Pseudonorms determine Stein morphism structures.

Extension theorem variant aids in proofs.

Abstract

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.