Oka's lemma

For Oka's lemma about coherent sheaves, see Oka coherence theorem.

In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in ${\displaystyle \mathbb {C} ^{n}}$, the function ${\displaystyle -\log d(z)}$ is plurisubharmonic, where ${\displaystyle d}$ is the distance to the boundary. This property shows that the domain is pseudoconvex. Historically, this lemma was first shown in the Hartogs domain in the case of two variables. Furthermore, Oka's lemma is the inverse of Levi's problem (unramified Riemann domain over ${\displaystyle \mathbb {C} ^{n}}$). Perhaps, this is why Oka referred to Levi's problem as "problème inverse de Hartogs", and could explain why Levi's problem is occasionally referred to as Hartogs' Inverse Problem.

References

• Harrington, Phillip S. (2007), "A quantitative analysis of Oka's lemma", Mathematische Zeitschrift, 256 (1): 113–138, doi:10.1007/s00209-006-0062-7, MR 2282262, S2CID 121735220
• Harrington, Phillip S.; Shaw, Mei-Chi (2007), "The strong Oka's lemma, bounded plurisubharmonic functions and the ${\displaystyle {\overline {\partial }}}$-Neumann problem", Asian Journal of Mathematics, 11 (1): 127–139, doi:10.4310/AJM.2007.v11.n1.a12, MR 2304586
• Herbig, A.-K.; McNeal, J. D. (2012), "Oka's lemma, convexity, and intermediate positivity conditions", Illinois Journal of Mathematics, 56 (1): 195–211 (2013), arXiv:1112.5138, doi:10.1215/ijm/1380287467, MR 3117025, S2CID 118437110
• Oka, Kiyoshi (1953), "Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur", Japanese Journal of Mathematics, 23: 97–155 (1954), doi:10.4099/jjm1924.23.0_97, MR 0071089
• Siu, Yum-Tong (1978), "Pseudoconvexity and the problem of Levi", Bulletin of the American Mathematical Society, 84 (4): 481–513, doi:10.1090/S0002-9904-1978-14483-8

Further reading

• Noguchi, Junjiro (2019). "A brief chronicle of the Levi (Hartog's inverse) problem, coherence and open problem". Notices of the International Congress of Chinese Mathematicians. 7 (2): 19–24. arXiv:1807.08246. doi:10.4310/ICCM.2019.V7.N2.A2. S2CID 119619733.
• Oka, Kiyoshi (1953), "Domaines finis sans point critique intérieur", Japanese Journal of Mathematics, 27: 97–155, doi:10.4099/jjm1924.23.0_97 PDF TeX