# Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle?

A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don’t really have time to learn all basics of sheaves theory and the connection to vector bundles. I just need a definition that uses the terms that come with the definition of vector bundle, so that I can check if the conditions are satisfied in my cases.

I am sorry if this seems a lazy questions. Of course I appreciate an answer that, in addition, lets me understand the connection.

#### Solutions Collecting From Web of "Globally generated vector bundle"

A holomorphic vector bundle $E$ is globally generated if there exist holomorphic sections $s_1, \dots, s_r$ such that for all $x \in X$, $s_1(x), \dots, s_r(x)$ span $E_x$.

In terms of the sheaf-theoretic notion you came across, a holomorphic vector bundle $E$ is globally generated if $\mathcal{O}(E)$, its sheaf of holomorphic sections, is globally generated as a sheaf.