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 $L\_i$. 



Can we patch up line bundles?
The cocycle condition for glueing applies to sheaves on any topological space, in particular to line bundles. See for instance Proposition 5.29 of


<http://math.rice.edu/~hassett/teaching/465spring04/CCAGlec5.pdf>.


