A brief chronicle of the Levi (Hartog’s inverse) problem, coherence and open problem

Here we chronologically summarize briefly the developments of the Levi (Hartogs’ Inverse) Problem together with the notion of coherence and its solution, shedding light on some records which have not been discussed in the past references. In particular, we will discuss K. Oka’s unpublished papers 1943 which solved the Levi (Hartogs’ Inverse) Problem for unramified Riemann domains of arbitrary dimension $n \geq 2$, usually referred as it was solved by Oka IX in 1953, H.J. Bremermann and F. Norguet in 1954 for univalent domains, independently.

At the end we emphasize an open problem in a ramified case.

Keywords

coherence, Oka, Levi problem, several complex variables

2010 Mathematics Subject Classification

32A99, 32E30

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Dedicated to the memory of Professor Akira Takeuchi.

Research supported in part by Grant-in-Aid for Scientific Research (C) 15K04917.

Published 8 August 2019