  In this work we prove that any unitary Sobolev $W^{1,2}$ connection of an
Hermitian bundle over a 2-dimensional K\"ahler manifold whose curvature is
$(1,1)$ defines a smooth holomorphic structure. We prove moreover that such a
connection can be strongly approximated in any $W^{1,p}$ ($p<2$) norm by smooth
connections satisfying the same integrability condition.
