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 $L_1^2$ of $L^2$ sections with $L^2$ first order derivatives in the sense of distributions, which satisfies furthermore $(\mathrm{Id}-\pi )\circ D^{\prime \prime }\pi =0$. We give a new simple proof of the fact that a weakly holomorphic subbundle of $(E,h)$ defines a coherent subsheaf of $\mathcal{O}(E),$ that is a holomorphic subbundle of $E$ in the complement of an analytic set of codimension $\ge 2.$ This result was the crucial technical argument in Uhlenbeck's and Yau's proof of the Kobayashi-Hitchin correspondence on compact Kähler 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.

19 pages