Suppose I have a trivial rank $k$ bundle $E$ over $\mathbb C^n$. Suppose that on $\mathbb C^n\setminus 0$ I have two algebraic sub-bundles $V\_1,V\_2\subset E$ of ranks $l$ and $k-l$ such that $V\_1\oplus V\_2=E$. Is it possible then to extend $V\_1$ and $V\_2$ as sub-bundles of $E$ to the whole $\mathbb C^n$? If not, what would be a counterexample?



Splitting a trivial bundle over punctured $\mathbb C^n$
The decomposition $\mathscr E|\_U = V\_1 \oplus V\_2$ gives projectors $\pi\_i \colon \mathscr E|\_U \to \mathscr E|\_U$ such that $\operatorname{im}(\pi\_i) = V\_i$. By Hartog's lemma, the restriction $\Gamma(X,\mathscr End(\mathscr E)) \to \Gamma(U,\mathscr End(\mathscr E))$ is a ring isomorphism, so $\pi\_i$ extend to projectors $\mathscr E \to \mathscr E$ with $\pi\_1 + \pi\_2 = 1$, which gives a decomposition of $\mathscr E$ restricting to the given one on $\mathscr E|\_U$. $\square$


