Holomorphic vector bundle

Remarks

Given two such trivializations $\varphi_i$ over $U_i$ and $\varphi_j$ over $U_j$, one has $\varphi_i \circ \varphi_j^{-1}(x, v) = (x, g_{ij}(x) v)$ with $g_{ij} \colon U_i \cap U_j \to GL_r(\C)$ holomorphic. Conversely, holomorphic maps $g_{ij}$ with $g_{ij} g_{jk} = g_{ik}$ on triple overlaps glue the products $U_i \times \C^r$ to a holomorphic vector bundle. A holomorphic vector bundle is in particular a smooth complex vector bundle.

Example: the holomorphic tangent bundle $T X$ of a complex manifold of dimension $n$ has rank $n$. A holomorphic chart $z = (z_1, \dots, z_n)$ trivializes it by the frame $\partial/\partial z_1, \dots, \partial/\partial z_n$. The transition functions are the complex Jacobian matrices of the coordinate changes, which are holomorphic.

Non-example: cover the Riemann sphere by $U_0 = \C$ and $U_1 = \mathbb{P}^1 \setminus \set{0}$, and glue $U_0 \times \C$ and $U_1 \times \C$ over $\C^*$ with $g_{01}(z) = \bar z$. This gives a smooth complex line bundle, but $g_{01}$ is not holomorphic. So the trivializations of this atlas are not biholomorphic for any complex structure on the glued space, and clause 2 fails for them.

The sheaf of holomorphic sections of $E$ is a locally free sheaf of $\mathcal{O}_X$-modules of rank $r$, and every such sheaf arises in this way.