A Simple Proof of a Theorem by Uhlenbeck and Yau

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 K\"ahler 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.