2011/08/10 by Junjirō Noguchi, Junjiro Noguchi, Noguchi, Junjiro
Mathematics · #32E40 #32T05 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.CV #msc:32E40 #msc:32T05
paper · pdf · doi:10.48550/arxiv.1108.2078
arxiv created 2011/08/10 · openalex publication_date 2011/08/10 · arxiv updated 2011/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \Cn 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 Hörmander'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.