Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Section Extension from Hyperbolic Geometry of Punctured Disk and
  Holomorphic Family of Flat Bundles   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 d-bar equation with cut-off functions and additional
blowup weight functions.

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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th