Classifying map for the hyperplane bundle

Question

Let $G$ be a topological group and $X$ be a CW complex. Then principal $G$-bundles on $X$ is classified by the classifying space $BG$ in the sense that, given a principal $G$-bundle $P \to X$, there is a continuous map $f\colon X \to BG$ such that $P = f^* EG$, where $EG$ is a contractible space on which $G$ acts freely with quotient $BG$. In the case when $G = S^1$, it is known that $BG = \mathbb{C}P^{\infty}$.

If we state this in terms of associated comlex line bundles with the standard $S^1$-action on $\mathbb{C}$, given a complex line bundle $L \to X$, there is a map $f\colon X \to \mathbb{C}P^{\infty}$ such that $L$ is the pullback of the universal line bundle by $f$.

When $X = \mathbb{C}P^n$, there are two important line bundles on $X$. The first one is the tautological line bundle, usually denoted by $\mathcal{O}(-1)$, which is defined to be the bundle whose fiber at $\ell \in \mathbb{C}P^n$ consists of points on $\ell$ considered as a line in $\mathbb{C}^{n+1}$. In this case, the classifying map $f\colon \mathbb{C}P^n \to \mathbb{C}P^{\infty}$ is the natural inclusion. The other one is the hyperplane bundle, the dual bundle of the tautological one. My question is

What is the classifying map for the hyperplane bundle $\mathcal{O}(1) \to \mathbb{C}P^n$? More generally, can we find the classifying map for $\mathcal{O}(k)$ for each $k$?

If this is difficult or complicated, the formula for $n=1$ or any reference would be appreciated.

Answer 1

If the coordinates change of a line bundle is $u_{ij}$, the coordinate change of the dual are $u_{tij}^{-1}$, for $n=1$, the map of $\mathbb{C}P^1$ induced by $(z_1,z_2)\rightarrow (\bar z_1,\bar z_2)$ change $O(1)$ to $O(-1)$. Just see how it acts on the coordinates change.

Answer 2

For $k > 0$, $\mathcal{O}(k) \to \mathbb{CP}^n$ is very ample, so there is an embedding $\varphi : \mathbb{CP}^n \to \mathbb{CP}^N$ such that $\varphi^*\mathcal{O}(1) \cong \mathcal{O}(k)$ (and hence $\varphi^*\mathcal{O}(-1) \cong \mathcal{O}(-k)$).

As the other answer alludes to, complex conjugation $c : \mathbb{C}^{n+1} \to \mathbb{C}^{n+1}$ induces a map $c : \mathbb{CP}^n \to \mathbb{CP}^n$ and if $E$ is a complex vector bundle, $c^*E \cong \overline{E}\cong E^*$, so $c^*\mathcal{O}(-k) \cong \mathcal{O}(k)$.

Let $i$ denote the inclusion $\mathbb{CP}^N \hookrightarrow \mathbb{CP}^{\infty}$. Combining the two facts above, we see that $i\circ\varphi$ is a classifying map for $\mathcal{O}(-k) \to \mathbb{CP}^n$, and $i\circ c\circ \varphi$ is a classifying map for $\mathcal{O}(k) \to \mathbb{CP}^n$ (as is $i\circ\varphi\circ c$).