Another Direct Proof of Oka's Theorem (Oka IX)

TL;DR

This paper presents a new, direct elementary proof of Oka's Theorem, solving Levi's problem for Riemann domains over complex n-space, using Grauert's finiteness theorem, induction on dimension, and jets.

Contribution

It offers a novel, straightforward proof of Oka's Theorem that simplifies previous approaches by relying solely on Grauert's finiteness theorem and inductive techniques.

Findings

Provides a comprehensive, elementary proof of Oka's Theorem.

Demonstrates the effectiveness of induction on dimension in complex analysis.

Simplifies understanding of Levi's problem for Riemann domains.

Abstract

In 1953 K. Oka IX solved in first and in a final form Levi's problem (Hartogs' inverse problem) for domains or Riemann domains over $\C^n$ of arbitrary dimension. Later on a number of the proofs were given; cf.\ e.g., Docquier-Grauert's paper in 1960, R. Narasimhan's paper in 1961/62, Gunning-Rossi's book, and H\"ormander's book (in which the holomorphic separability is pre-assumed in the definition of Riemann domains and thus the assumption is stronger than the one in the present paper). Here we will give another direct elementary proof of Oka's Theorem, relying only on Grauert's finiteness theorem by the {\it induction on the dimension} and the {\it jets over Riemann domains}; hopefully, the proof is most comprehensive.