Let $X$ be a complex manifold, let $r \in \mathbb{N}$, let $I_r \in GL(r,\mathbb{C})$ denote the $r \times r$ identity matrix, and let $E \to X$ be a holomorphic vector bundle of rank $r$. Let $(U_i)_{i \in I}$ be a holomorphic trivializing open cover of $X$, and let
be the transition functions of $E$ with the convention that, for local holomorphic frames $e_i = (e_{i,1},\dots,e_{i,r})$ and $e_j = (e_{j,1},\dots,e_{j,r})$, one has $e_j = e_i g_{ij}$ on $U_i \cap U_j$. Define
by $h_{ij}(x) = \det(g_{ij}(x))$. Then the functions $(h_{ij})_{i,j \in I}$ form a holomorphic line-bundle cocycle. The holomorphic line bundle defined by this cocycle is canonically isomorphic to the top exterior power $\Lambda^r E$, and is denoted $\det E$.