Confused on Projective Bundle

Question

I am working through Atiyah (https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf) and am confused on the projective bundle (pg. 44-45). Atiyah states that for a vector bundle $E$ over $X$, let $E_0 = E-X$ and then $P(E)$ or the projective bundle is just the quotient space of $E_0$ divided by the action of the complex numbers. My confusion is twofold: how can one subtract $X$ from $E$, or what does Atiyah mean by that? Also, I am a bit confused on what the "action of the complex numbers" means. I have tried looking it up and haven't found anything helpful. For context, I am working through this book as part of a program at my university without having taken algebra or topology properly yet, so I am just trying to get as good of a grasp as possible. Any help would be greatly appreciated!

Answer 1

As pointed out in the comments, $E - X$ is another notation for $E\setminus X$. As Atiyah mentions, here we are viewing $X$ inside of $E$ via the zero section.

As $E \to X$ is a complex vector bundle, the fibers are complex vector spaces, so $\mathbb{C}$ acts on $E$ via scalar multiplication. Removing the zero section, we see that $\mathbb{C}^*$ acts on $E - X$ via scalar multiplication. Then $\mathbb{P}(E) = (E-X)/\mathbb{C}^*$.