Let $M$ be a complex manifold (of dimension greater than one), $L$ a line bundle, and $\Delta$ a connection on $L$ with possibly singularities along a divisor $D$. We define the curvature as $$ R_{\Delta}(X,Y)=[\Delta_X,\Delta_Y]-\Delta_{[X,Y]} $$

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

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 $\nabla'$?