A Simple Proof of a Theorem by Uhlenbeck and Yau

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 π lying in the Sobolev space L1 of L 2 sections with L2 first order derivatives in the sense of distributions, which satisfies furthermore (Id − π) ◦ D′′π = 0. We give a new simple proof of the fact that a weakly holomorphic subbundle of (E, h) defines a coherent subsheaf of O(E), that is a holomorphic subbundle of E in the complement of an analytic set of codimension ≥ 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 Imπ 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. 0.