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A simple proof of a theorem by Uhlenbeck and Yau
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UNSPECIFIED (2005) A simple proof of a theorem by Uhlenbeck and Yau. MATHEMATISCHE ZEITSCHRIFT, 250 (4). pp. 855-872. doi:10.1007/s00209-005-0780-2 ISSN 0025-5874.
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Official URL: http://dx.doi.org/10.1007/s00209-005-0780-2
Abstract
A subbundle of a Hermitian holomorphic vector bundle (E, h) can be metrically and differentially defined by the orthogonal projection onto itself. A weakly holomorphic subbundle of (E, h) is, by definition, an orthogonal projection p lying in the Sobolev space L-1(2) of L-2 sections of End E with L-2 first order derivatives in the sense of distributions, which satisfies furthermore (Id-pi) circle D '' pi = 0. A weakly holomorphic subbundle of (E, h) is shown to define a coherent subsheaf of O(E), and implicitly a holomorphic subbundle of E in the complement of an analytic subset of codimension >= 2. This result provided the key technical argument to the proof given by Uhlenbeck and Yau for the Kobayashi-Hitchin correspondence on compact Kahler manifolds. We give here a much simpler proof based on current theory. The idea is to construct local meromorphic sections of Im pi which locally span the fibers. We first make this construction on one-dimensional sub-manifolds of X and subsequently extend it by means of a Hartogs-type theorem of Shiffman's.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Journal or Publication Title: | MATHEMATISCHE ZEITSCHRIFT | ||||
Publisher: | SPRINGER | ||||
ISSN: | 0025-5874 | ||||
Official Date: | August 2005 | ||||
Dates: |
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Volume: | 250 | ||||
Number: | 4 | ||||
Number of Pages: | 18 | ||||
Page Range: | pp. 855-872 | ||||
DOI: | 10.1007/s00209-005-0780-2 | ||||
Publication Status: | Published |
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