Ohsawa–Takegoshi L2 extension theorem

In several complex variables, the Ohsawa–Takegoshi L^2 extension theorem is a fundamental result concerning the holomorphic extension of an ${\displaystyle L^{2}}$-holomorphic function defined on a bounded Stein manifold (such as a pseudoconvex compact set in ${\displaystyle \mathbb {C} ^{n}}$ of dimension less than ${\displaystyle n}$) to a domain of higher dimension, with a bound on the growth. It was discovered by Takeo Ohsawa and Kensho Takegoshi in 1987,[1] using what have been described as ad hoc methods involving twisted Laplace–Beltrami operators, but simpler proofs have since been discovered.[2] Many generalizations and similar results exist, and are known as theorems of Ohsawa–Takegoshi type.

See also

• Suita conjecture

note

1. Ohsawa & Takegoshi (1987)
2. Siu (2011)

References

• Błocki, Zbigniew (2014). "Cauchy–Riemann meet Monge–Ampère". Bulletin of Mathematical Sciences. 4 (3): 433–480. doi:10.1007/s13373-014-0058-2. S2CID 53582451.
• Demailly, Jean-Pierre (2000). "On the Ohsawa–Takegoshi–Manivel L^2 extension theorem" (PDF). Complex Analysis and Geometry. Progress in Mathematics. Vol. 188. pp. 47–82. doi:10.1007/978-3-0348-8436-5_3. ISBN 978-3-0348-9566-8.
• Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an ${\displaystyle L^{2}}$ extension problem with an optimal estimate and applications". Annals of Mathematics. 181 (3): 1139–1208. arXiv:1310.7169. doi:10.4007/annals.2015.181.3.6. JSTOR 24523356. S2CID 56205818.
• Hörmander, Lars (1965). "L^2 estimates and existence theorems for the ${\displaystyle {\bar {\partial }}}$ operator". Acta Mathematica. 113: 89–152. doi:10.1007/BF02391775. S2CID 120051843.
• Ohsawa, Takeo; Takegoshi, Kensho (1987). "On the extension of ${\displaystyle L^{2}}$ holomorphic functions". Mathematische Zeitschrift. 195 (2): 197–204. doi:10.1007/BF01166457. S2CID 122156071.
• Ohsawa, Takeo (2017). "On the extension of ${\displaystyle L^{2}}$ holomorphic functions VIII — a remark on a theorem of Guan and Zhou". International Journal of Mathematics. 28 (9). doi:10.1142/S0129167X17400055.
• Ohsawa, Takeo (10 December 2018). ${\displaystyle L^{2}}$ Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds. Springer Monographs in Mathematics. doi:10.1007/978-4-431-55747-0. ISBN 9784431568513.
• Bousfield Classes and Ohkawa's Theorem. Springer Proceedings in Mathematics & Statistics. Vol. 309. 2020. doi:10.1007/978-981-15-1588-0. ISBN 978-981-15-1587-3. S2CID 242194764.
• Siu, Yum-Tong (2011). "Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles". Science China Mathematics. 54 (8): 1767–1802. arXiv:1104.2563. Bibcode:2011ScChA..54.1767S. doi:10.1007/s11425-011-4293-7. S2CID 119572640.

External links

• Demailly, Jean-Pierre (June 1996). "L^2 estimates for the d-bar operator on complex manifolds, Notes de cours, Ecole d'été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble" (PDF).
• Analytic Methods in Algebraic Geometry (OpenContent book See B5)