Let $\left(E,h\right)$ be a hermitian holomorphic vector bundle over a complex manifold $M$. Then there exists a unique hermitian connection $\nabla^h$, the $\left(0,1\right)$-part of which is the holomoprhic structure of $E$.
If one assumes that the curvature of $\nabla^h$ is square integrable then a number of interesting theorems can be proven. Such as extending the bundle $E$ across an analytic subset $A$ as reflexive sheaves, when $M= N\setminus A$ (e.g. discussed by V. Shevchishin).
The reason that integrability allows such results is clearly of an analytic nature, since mostly the integrability is used to solve certian PDEs.
My rather vague question is: Can integrable Chern Curvature be described in more Algebro-Geometric terms?
Clearly it implies a non zero amount about the underlying bundle and the extension results suggest some type of "triviality" towards "infinity" (e.g. assume $N$ is a compactification of $M$).
Essentially I am interested in results that reformulate the integrablity condition into other terms or implications of integrable curvature.