A simple proof of a theorem by Uhlenbeck and Yau

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 π lying in the Sobolev space L21 of L2 sections of End E with L2 first order derivatives in the sense of distributions, which satisfies furthermore (Id−π)∘D′′π=0. A weakly holomorphic subbundle of (E, h) is shown to define a coherent subsheaf of (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 Kähler manifolds. We give here a much simpler proof based on current theory. The idea is to construct local meromorphic sections of Im π which locally span the fibers. We first make this construction on one-dimensional submanifolds of X and subsequently extend it by means of a Hartogs-type theorem of Shiffman’s.