The [Quillen-Suslin][1] theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector bundles on the *punctured* affine space $X_n=\mathbb{A}^{n+1}\smallsetminus\{0\}$?
According to [this paper][2], there seem to be room for nontrivial vector bundles.

> **1.** Is there a classification of vector bundles on $X_n$, at least for some specific ranks? 

Now let $\mathbb{C}^{*}$ act on $X_n$ by the action $\lambda.(x_0,\dots,x_n):=(\lambda x_0,\lambda x_1,\dots, \lambda x_n)$ whose quotient is $\mathbb{P}^n$.

> **2.** What about $\mathbb{C}^{*}$-equivariant vector bundles on $X_n$? Does the presence of an equivariant structure restrict the possibilities for (the isomorphism class of) a vector bundle on $X_n$? 

> In particular, are all equivariant v.b. on $X_n$ necessarily trivial (as algebraic v.b.)?(Notice that, since equivariant v.b. on $X_n$ are in bijection -via pullback- with v.b. on $\mathbb{P}^n$, the last question amounts to asking whether all pullback v.b. fom $\mathbb{P}^n$ are trivial on $X_n$).






  [1]: https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem
  [2]: https://arxiv.org/pdf/1303.0575.pdf