Let $X$ be a toric variety, and $\pi:E\to X$ a vector bundle, say of rank $2$. You can think of $X=\mathbb P^1$.
When is the total space of $E$, or of $P(E)$, a toric variety? What do I need in order to lift the torus action on $X$ to a torus action on $E$, or on $P(E)$, so to get an open orbit? I ask this because I was reading this thesis, and on page $2$ I found the following statement, which I do not understand:
If a vector bundle over a toric variety splits as a sum of line bundles, then its projectivization admits a toric variety structure".
I want to understand at least the case of $\mathbb P^1$, where every vector bundle splits. But I have no idea why the splitting is so relevant in general.