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On Oka’s extra-zero problem and examples

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Abstract

After the solution of Cousin II problem by Oka III in 1939, he thought an extra-zero problem in 1945 (his posthumous paper) asking if it is possible to solve an arbitrarily given Cousin II problem adding some extra-zeros whose support is disjoint from the given one. By the secondly named author, some special case was affirmatively confirmed in dimension two and a counter-example in dimension three or more was given. The purpose of the present paper is to give a complete solution of this problem with examples and some new questions.

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Notes

  1. Here his term is “balayable” used in Oka [12] the meaning is that the given Cousin II distribution is continuously deformable to a zero-free continuous Cousin II distribution. The Cousin II problem on a domain \(X\) of holomorphy is solvable if and only if \(D\) is balayable on \(X\).

  2. He did not give an explicit problem here.

  3. It is now very difficult to find a complete correct record of Oka’s publications without errors. The most commonly referred reference of Oka’s works is “Kiyoshi Oka Collected Papers”, translated by Narasimhan, R., with commentaries by Cartan, H., edited by Remmert, R., Springer, Berlin (1984), where there are some mistakes in the records and moreover all records of the received dates were deleted. Therefore the authors think that it is meaningful and useful to provide a complete list of his publications with the received dates in one place (Oka [10–22]).

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Acknowledgments

After the counter-example constructed by Hamano [6] which was a reducible divisor, Professor T. Ueda asked if there is an irreducible counter-example; his question forms a part of the motivation of the present paper. Professor S. Takayama gave an interesting example of Sect. 4. The authors are very grateful to all of them.

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Authors and Affiliations

  1. Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, 1-7-1 Kagamiyama, Higashi-Hiroshima, 739-8521, Japan

    Makoto Abe

  2. Department of Mathematics, Faculty of Human Development and Culture, Fukushima University, Fukushima, Kanayagawa, 960-1296 , Japan

    Sachiko Hamano

  3. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Meguroku, Komaba, 153-8914, Japan

    Junjiro Noguchi

Authors
  1. Makoto Abe
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  2. Sachiko Hamano
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  3. Junjiro Noguchi
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Correspondence to Junjiro Noguchi.

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Research supported in part by Grants-in-Aid for Scientific Research (C) 23540217, for Young Scientists (B) 23740098 and for Scientific Research (B) 23340029 of Japan Society for the Promotion of Science.

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Abe, M., Hamano, S. & Noguchi, J. On Oka’s extra-zero problem and examples. Math. Z. 275, 79–89 (2013). https://doi.org/10.1007/s00209-012-1123-8

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