Let $X$ be a complex manifold, let $r \in \mathbb{N}$, and let $(U_i)_{i \in I}$ be an open cover of $X$. Suppose that, for every $i,j \in I$, there is a holomorphic map
Then the collection $(g_{ij})_{i,j \in I}$ defines a holomorphic vector bundle of rank $r$ over $X$ whose transition functions with respect to the cover $(U_i)_{i \in I}$ are the maps $g_{ij}$.
paragraph
admin
Moreover, if $(g'_{ij})_{i,j \in I}$ is another such cocycle on the same cover, then the holomorphic vector bundles defined by $(g_{ij})$ and $(g'_{ij})$ are isomorphic if and only if there exist holomorphic maps
If two cocycles are written on different open covers, they are compared by pulling both cocycles back to a common open refinement and applying the preceding criterion.