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

J. Math. Sci. Univ. Tokyo
Vol. 19 (2012), No. 4, Page 661–675.

Noguchi, Junjiro
Another Direct Proof of Oka's Theorem (Oka IX)
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}; here we do {\em not} use even Behnke-Stein's theorem on the Steiness of an open Riemann surface. Hopefully, the proof is the easiest.