# Weak Holomorphic Structures over Kähler Surfaces

@article{Punoiu2019WeakHS, title={Weak Holomorphic Structures over K{\"a}hler Surfaces}, author={Alexandru Păunoiu and Tristan Rivi{\'e}re}, journal={arXiv: Differential Geometry}, year={2019} }

In this work we prove that any unitary Sobolev $W^{1,2}$ connection of an Hermitian bundle over a 2-dimensional Kahler 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.

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