Holomorphic Frames for Weakly Converging Holomorphic Vector Bundles

G. Daskalopoulos and R. Wentworth

This is a technical note on the Newlander-Nirenberg Theorem in the context of integrable d-bar operators on complex vector bundles. We show, using the technique of Webster, that local holomorphic frames can be constructed in a uniform way for Holder continuous d-bar operators. As a consequence, convergence of holomorphic frames can be achieved for operators converging weakly in L^p_1 for p larger than the dimension of the manifold. Such convergence occurs often in the context of Uhlenbeck compactness.