## A TAKAYAMA-TYPE EXTENSION THEOREM (2006)

### BibTeX

@MISC{06atakayama-type,

author = {},

title = {A TAKAYAMA-TYPE EXTENSION THEOREM},

year = {2006}

}

### OpenURL

### Abstract

Let X be a compact complex algebraic manifold, E → X a holomorphic line bundle, and Z ⊂ X a smooth codimension-1 submanifold. The main goal of this paper is to establish sufficient conditions for extending sections of the pluri-adjoint bundles m(KZ + E|Z) from Z to X.

### Citations

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(Show Context)
Citation Context ...ections of the line bundle associated to L. For a more thorough discussion of this and many other matters in algebraic geometry, as well as a more algebro-geometric approach to multiplier ideals, see =-=[L-04]-=-. Theorem 2.3. Let X be a projective algebraic manifold, Z ⊂ X a smooth divisor, and E an integral divisor on X . Assume that one can write E ∼Q E1 + E2 such that the following properties hold. (Pa) F... |

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Citation Context ...n the work of Ohsawa and Takegoshi, in the context of extension of holomorphic functions, square integrable with respect to a plurisubharmonic weight, from hyperplanes to a pseudoconvex domain in C n =-=[OT-87]-=-. Since that time there have been several extensions of the result. The simplest general exposition was finally given by Siu [S-02], and used as one of two fundamental tools in establishing the celebr... |

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(Show Context)
Citation Context ...ubharmonic weight, from hyperplanes to a pseudoconvex domain in C n [OT-87]. Since that time there have been several extensions of the result. The simplest general exposition was finally given by Siu =-=[S-02]-=-, and used as one of two fundamental tools in establishing the celebrated deformation invariance of plurigenera. Extensions to more singular settings were established by the author and McNeal in [MV-0... |

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(Show Context)
Citation Context ...integer m > 0 such that mD is a Z-divisor and sm ∈ H0 (X, mD).)TAKAYAMA EXTENSION 5 Theorem 2.1 is a corollary of Theorem 1. To see this, we argue as follows. It is not hard to see (cf. Section 4 in =-=[T-05]-=-) that we may assume without loss of generality that A is an ample Q-divisor. Thus we can construct a singular metric e−ϕEof positive curvature for E as follows: take a multiple mA that is very ample,... |

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(Show Context)
Citation Context ... to eliminate this positive line bundle, through the method of taking powers and roots. Our construction of the metric e −ψ is substantially shortened by the use of a new method of Paun introduced in =-=[P-05]-=-. Paun has eliminated the need for the use of an effective global generation of multiplier ideal sheaves, which was a long and difficult part of Siu’s approach. Finally we should mention the recent pr... |

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(Show Context)
Citation Context ...S-02], and used as one of two fundamental tools in establishing the celebrated deformation invariance of plurigenera. Extensions to more singular settings were established by the author and McNeal in =-=[MV-05]-=-. The positivity assumptions in Theorem 1 are in some sense minimal, and in particular are insufficient to support the use of vanishing theorems. A technique for dealing with such a situation was init... |

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(Show Context)
Citation Context ...has eliminated the need for the use of an effective global generation of multiplier ideal sheaves, which was a long and difficult part of Siu’s approach. Finally we should mention the recent preprint =-=[C]-=- of Claudon, who handles the case where the normal bundle of Z is trivial. (Claudon has assumed that the family4 DROR VAROLIN lies over a disk.) Claudon’s result is a direct adaptation of Siu’s (or m... |

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