Let $M$ be a simply connected complex manifold (of dimension greater than one), $L$ a line bundle, and $\nabla$ a connection on $L$ with possibly singularities along a divisor $D$. We define the curvature as
$$
R\_{\nabla}(X,Y)=[\nabla\_X,\nabla\_Y]-\nabla\_{[X,Y]}
$$


Suppose that a second connection $\nabla'$ on $L$ has the same singularities and the same curvature. Is $\nabla$ equal to $\nabla'$?



Does the holomorphic curvature determine the connection?
The connection can be equivalently given by parallel transport along curves. So if two connections are equal (or isomorphic in appropriate sense) then they must have the same holonomy. Now by the Ambrose--Singer theorem there is a close relationship between holonomy and curvature. But full holonomy group is a global object and can see holes ($\pi\_1(M)$) in your manifold. As has been already mentioned in the comments, point singularities of connections pretty much behave as punctures in $M$.


