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- W2949456644 abstract "A subbundle of a Hermitian vector bundle $(E, h)$ can be metrically and differentiably defined by the orthogonal projection onto this subbundle. A weakly holomorphic subbundle of a Hermitian holomorphic bundle is, by definition, an orthogonal projection $pi$ lying in the Sobolev space $L^2_1$ of $L^2$ sections with $L^2$ first order derivatives in the sense of distributions, which satisfies furthermore $(mathrm{Id}-pi)circ D''pi=0$. We give a new simple proof of the fact that a weakly holomorphic subbundle of $(E, h)$ defines a coherent subsheaf of ${cal O}(E),$ that is a holomorphic subbundle of $E$ in the complement of an analytic set of codimension $geq 2.$ This result was the crucial technical argument in Uhlenbeck's and Yau's proof of 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 $mathrm{Im} pi$ which locally span the fibers. We first make this construction on every one-dimensional submanifold of $X$ and subsequently extend it via a Hartogs-type theorem of Shiffman's." @default.
- W2949456644 created "2019-06-27" @default.
- W2949456644 creator A5028699377 @default.
- W2949456644 date "2005-01-01" @default.
- W2949456644 modified "2023-09-23" @default.
- W2949456644 title "A Simple Proof of a Theorem by Uhlenbeck and Yau" @default.
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