Ohsawa–Takegoshi L2 extension theorem

Short description: Result concerning the holomorphic extensions In several complex variables

In several complex variables, the Ohsawa–Takegoshi L² extension theorem is a fundamental result concerning the holomorphic extension of an [math]\displaystyle{ L^2 }[/math]-holomorphic function defined on a bounded Stein manifold (such as a pseudoconvex compact set in [math]\displaystyle{ \mathbb{C}^n }[/math] of dimension less than [math]\displaystyle{ n }[/math]) to a domain of higher dimension, with a bound on the growth. It was discovered by Takeo Ohsawa and Kensho Takegoshi in 1987, using what have been described as ad hoc methods involving twisted Laplace–Beltrami operators, but simpler proofs have since been discovered. Many generalizations and similar results exist, and are known as theorems of Ohsawa–Takegoshi type.

See also

• Suita conjecture

References

• Błocki, Zbigniew (2014). "Cauchy–Riemann meet Monge–Ampère". Bulletin of Mathematical Sciences 4 (3): 433–480. doi:10.1007/s13373-014-0058-2. 
• Demailly, Jean-Pierre (2000). "On the Ohsawa–Takegoshi–Manivel L² extension theorem". Complex Analysis and Geometry. pp. 47–82. doi:10.1007/978-3-0348-8436-5_3. ISBN 978-3-0348-9566-8. https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/ohsawa_tak.pdf. 
• Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an [math]\displaystyle{ L^2 }[/math] extension problem with an optimal estimate and applications". Annals of Mathematics 181 (3): 1139–1208. doi:10.4007/annals.2015.181.3.6. 
• Hörmander, Lars (1965). "L² estimates and existence theorems for the [math]\displaystyle{ \bar \partial }[/math] operator". Acta Mathematica 113: 89–152. doi:10.1007/BF02391775. 
• Ohsawa, Takeo; Takegoshi, Kensho (1987). "On the extension of [math]\displaystyle{ L^2 }[/math] holomorphic functions". Mathematische Zeitschrift 195 (2): 197–204. doi:10.1007/BF01166457. 
• Ohsawa, Takeo (2017). "On the extension of [math]\displaystyle{ L^2 }[/math] 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). [math]\displaystyle{ L^2 }[/math] 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. 309. 2020. doi:10.1007/978-981-15-1588-0. ISBN 978-981-15-1587-3. https://books.google.com/books?id=ssPXDwAAQBAJ&pg=PA426. 
• 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. doi:10.1007/s11425-011-4293-7. Bibcode: 2011ScChA..54.1767S. 

External links

• Demailly, Jean-Pierre (June 1996). "L² estimates for the d-bar operator on complex manifolds, Notes de cours, Ecole d'été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble". https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/estimations_l2.pdf. 
• Analytic Methods in Algebraic Geometry (OpenContent book See B5)