I have two related questions. Can there be a stably trivial non-trivial holomorphic vector bundle over a closed complex manifold? Can there be a stably trivial non-trivial algebraic vector bundle over a smooth proper variety (of arbitary characteristic)?



Stably trivial non-trivial vector bundles
Assume $E \oplus \mathcal{O} \cong \mathcal{O}^{\oplus n}$. Then, of course, $E \cong \mathcal{O}^{\oplus n}/\mathcal{O}$. On the other hand
$$
Hom(\mathcal{O}, \mathcal{O}^{\oplus n}) \cong \Gamma(X, \mathcal{O})^{\oplus n}.
$$
If $X$ is proper, connected and reduced, then $\Gamma(X, \mathcal{O}) = \Bbbk$ (the base field), hence any non-zero morphism $\mathcal{O} \to \mathcal{O}^{\oplus n}$ is given by a non-zero $n$-tuple of elements of the field, therefore any such morphism is isomorphic to the embedding of the first direct summand, hence the quotient is isomorphic to $\mathcal{O}^{\oplus (n - 1)}$. Thus, $E$ is trivial.


