Are there any examples of holomorphic vector bundles $E \to M$ of rank $k$ that do not admit a holomorphic classifying map $M \to \mathrm{Gr}_k(\mathbb C^n)$ for any $n \in \mathbb N$? Or, equivalently, holomorphic vector bundles $E \to M$ that do not holomorphically embed in any free one.
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The line bundle $\mathcal{O}(1)$ does not embed into a trivial vector bundle, and in fact $\mathrm{Hom}(\mathcal{O}(1),\mathcal{O}) = 0$.
