Extension of Twisted Pluricanonical Sections with Plurisubharmonic Weight and Invariance of Semipositively Twisted Pluri
Let X be a holomorphic family of compact complex projective algebraic manifolds with fibers X t over the open unit 1-disk Δ. Let \( K_{x_t } \) and K X be respectively the canonical line bundles of X t and X. We prove that, if L is a holomorphic Une bundl
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Abstract Let X be a holomorphic family of compact complex projective algebraic manifolds with fibers X t over the open unit I-disk ..1. Let KX t and Kx be respectively the canonical line bundles of X t and X. We prove that, if L is a holomorphic line bundle over X with a (possibly singular) metric e-'f' of semipositive curvature current on X such that e-'f'Jxo is locally integrable on Xo, then for any positive integer m, any s E r(mK xo +L) with JSJ2 e -'f' locally bounded on Xo can be extended to an element of r(X,mKx + L). In particular, dimr (Xt, mKxt + L) is independent of t for cp smooth. The case of trivial L gives the deformational invariance of the plurigenera. The method of proof uses an appropriately formulated effective version, with estimates, of the argument in the author's earlier paper on the invariance of plurigenera for general type. A delicate point of the estimates involves the use of metrics as singular as possible for p K Xo + ap L on Xo to make the dimension of the space of L2 holomorphic sections over Xo bounded independently of p, where ap is the smallest integer ~ ~. These metrics are constructed from s. More conventional metrics, independent of s, such as generalized Bergman kernels are not singular enough for the estimates.
Table of Contents 1 2 3 4 5 6
Introduction ................................................ 224 Review of Existing Argument for Invariance of Plurigenera .. 228 Global Generation of Multiplier Ideal Sheaves with Estimates.234 Extension Theorems of Ohsawa-Takegoshi Type from Usual Basic Estimates with Two Weight Functions ...................... 241 Induction Argument with Estimates ......................... 248 Effective Version of the Process of Taking Powers and Roots of Sections .................................................... 256 Remarks on the Approach of Generalized Bergman Kernels. 264 References .................................................. 276
* Partially supported by a grant from the National Science Foundation. 2000 Mathematics Subject Classification: 32G05, 32C35. I. Bauer et al. (eds.), Complex Geometry © Springer-Verlag Berlin Heidelberg 2002
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Yum-Tong Siu
Introduction
For a holomorphic family of compact complex projective algebraic manifolds, the plurigenera of a fiber are conjectured to be independent of the fiber. The case when the fibers are of general type was proved in [Siu98]. Generalizations were made by Kawamata [Kaw99] and Nakayama [Nak98] and, in addition, they recast the transcendentally formulated methods in [Siu98] into a completely algebraic geometric setting. Recently Tsuji put on the web a preprint on the deformational invariance of the plurigenera for manifolds not necessarily of general type [TsuOl], in which, in addition to the techniques of [Siu98], he uses his theory of analytic Zariski decomposition and generalized Bergman kernels. Tsuji's approach of generalized Bergman kernels naturally reduces the problem of the deformational invariance of the plurigenera to a growth estimate on the generalized Bergman kerne
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