Let X be a complex manifold. Suppose we have holomorphic line bundles $L_i$ over $U_i$ where ${U_i}$ is an open covering of X. Suppose that $L_i$ and $L_j$ restrict to the same line bundle over the intersection of $U_i$ and $U_j$.
Can we patch these local line bundles into a global holomorphic line bundle L over X? That is, the restriction of L to $U_i$ is $L_i$.