Sobolev connections and holomorphic structures over Kähler surfaces

Abstract

In this work we prove that any unitary Sobolev W1,2 connection of an Hermitian bundle over a closed Kähler surface whose curvature is (1,1) defines a smooth holomorphic structure. We prove moreover that such a connection can be strongly approximated in any W1,p (p<2) norm by smooth connections satisfying the same integrability condition and consequently carrying smooth holomorphic structures.