Vector bundle over projective space

Question

The following exercise is from Atiyah's book "K-theory", Example 2.

Let $V$ be a vector space and let $\mathbb{P}(V)$ be its associated projective space. That is $\mathbb{P}(V)=\frac{V\backslash\{0\}}{\sim}$, where $\sim$ is the equivalence relation given by $v_1\sim v_2$ iff $\mathbb{C}v_1=\mathbb{C}v_2$. We define $E\subseteq \mathbb{P}(V)\times V$ to be the set of all $(x,v)$ such that $v$ lies in the line determining $x$. Show that $E$ is a vector bundle over $\mathbb{P}(V)$, with the canonical projection map $E\to \mathbb{P}(V)$ maps $(x,v)$ to $x$.

My question: First of all, I don't understand why $x$ determines a line, I mean- the equivalence calss $x=[v_1]$, seems to be a sphere, and not a line...because we consider complex scalars.

Secondly, I could not show that it is locally trivial. Given some $x=[v_1]$ in $\mathbb{P}(V)$, I need to find some open subset $U\subseteq \mathbb{P}(V)$ containing $x$, such that $E|_{U}$ is isomorphic to $U\times W$, for some vector space $W$. I have no idea how to do so... I noticed that the fivers are one-dimensional, for $x=[v_1]$, $E_x$ is isomorphic to the subspace spanned by $v_1$.

Thank you for any help!

Answer 1

$\require{AMScd}\def\CC{\mathbb{C}}$Let us assume that $V=\CC^{n+1}$. If $v\in V\setminus0$, let us write $[v]$ its class in $P(V)$.

The set $$U=\{[(v_0,\dots,v_n)]\in P(V):v_0\neq0\}$$ is an open set, and the map $\phi:\CC^n\to U$ such that $$\phi(x_1,\dots,x_n)=[(1,x_1,\dots,x_n)]$$ for all $(x_1,\dots,x_n)\in\CC^n$ is a homeomorphism.

Now let $E=P(V)\times V$ be the subset of al pairs $(p,v)$ of $P(V)\times V$ such that $v\in p$, and let $\pi:E\to P(V)$ be the restriction to $E$ of the map $(p,v)\in P(V)\times V\mapsto p\in P(V)$.

Let, on the other hand, $p:\CC^n\times\CC^n$ be the projection on the first factor, and let $\Phi:\CC^n\times\CC\to E$ be the map such that $$\Phi(x,\lambda)=(\phi(x),\lambda x'),$$ where if $x=(x_1,\dots,x_n)$ I am writing $x'=(1,x_1,\dots,x_n)$. The diagram \begin{CD} \CC^n\times\CC @>\Phi>> E \\ @VpVV @V\pi VV \\ \CC^n @>\phi>> P(V) \end{CD}

Now you have to check that this provides a trivialization of $E$ over the open set $U$, and use similar ideas to construct trivializations over all open sets of an open covering of $P(V)$.