Science China Mathematics

, 54:1767 | Cite as

Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles

  • Yum-Tong Siu


The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin’s application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the \(\bar \partial \) equation with cut-off functions and additional blowup weight functions.


\(\bar \partial \) estimate Ohsawa-Takegoshi extension hyperbolic geometry of punctured disk flatly twisted pluricanoincal section Gelfond-Schneider technique 


32210 14D20 


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© Science China Press and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA

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