Is there a relation between the complexification of the canonical real line bundle and the canonical complex line bundle. (which arises from the fact that any real vector bundle can be complexified)
Definitions required:
Given a real vector bundle $E$ over a topological space $X$, its complexification is defined to be the complex vector bundle $E\otimes_{\mathbb{R}}\mathbb{C}$ obtained by tensoring $E$ with the complex numbers $\mathbb{C}$ over the real numbers $\mathbb{R}$. In other words, the fibers of $E\otimes_{\mathbb{R}}\mathbb{C}$ are obtained by complexifying the fibers of $E$.
Now, the canonical real line bundle over the real projective space $\mathbb{RP}^n$ is defined as the quotient bundle $L\to\mathbb{RP}^n$ of the tautological bundle over the unit sphere $S^n\subseteq\mathbb{R}^{n+1}$, where $L$ is the subspace of $\mathbb{R}^{n+1}\times\mathbb{R}$ consisting of all pairs $(x,v)$ with $v\in\operatorname{span}{x}$.
What I thought:
The complexification of the canonical real line bundle over $\mathbb{RP}^n$ is given by the tensor product $L\otimes_{\mathbb{R}}\mathbb{C}$, which can be identified with the canonical complex line bundle over $\mathbb{CP}^n$, the complex projective space of complex dimension $n$. More precisely, the fibers of $L\otimes_{\mathbb{R}}\mathbb{C}$ over a point $[x]\in\mathbb{RP}^n$ can be identified with the complex line spanned by $x+0*i$ in $\mathbb{C}^{n+1}$, which is naturally isomorphic to the complex line $[x+0*i]\in\mathbb{CP}^n$.
In other words can we say that the complexification is isomorphic to the pullback of the complex canonical bundle via the inclusion?
Is my understanding of the subject correct, can you suggest any other reasons?
Thanks and regards in advance
Let $V$ be a real vector space and let $V_{\mathbb{C}}$ denote its complexification, i.e. $V_{\mathbb{C}} = V\otimes_{\mathbb{R}}\mathbb{C}$. Consider the natural inclusion $i : \mathbb{P}_{\mathbb{R}}(V) \to \mathbb{P}_{\mathbb{C}}(V_{\mathbb{C}})$ given by $\operatorname{span}_{\mathbb{R}}\{v\} \mapsto \operatorname{span}_{\mathbb{C}}\{v\otimes 1\}$. The total spaces of the tautological line bundles on $\mathbb{P}_{\mathbb{R}}(V)$ and $\mathbb{P}_{\mathbb{C}}(V_{\mathbb{C}})$ are
$$\gamma_{\mathbb{R}} = \{(\ell, v) \in \mathbb{P}_{\mathbb{R}}(V)\times V \mid v \in \ell\}$$
$$\gamma_{\mathbb{C}} = \{(L, w) \in \mathbb{P}_{\mathbb{C}}(V_{\mathbb{C}})\times V_{\mathbb{C}} \mid w \in L\}.$$
The total space of the pullback bundle $i^*\gamma_{\mathbb{C}} \to \mathbb{P}_{\mathbb{R}}(V)$ is given by
$$i^*\gamma_{\mathbb{C}} = \{(\ell, (L, w)) \in \mathbb{P}_{\mathbb{R}}(V)\times\gamma_{\mathbb{C}} \mid i(\ell) = L\}.$$
On the other hand, the complexified bundle $\gamma_{\mathbb{R}}\otimes\mathbb{C}$ is just
$$\gamma_{\mathbb{R}}\oplus\gamma_{\mathbb{R}} = \{((\ell_1, v_1), (\ell_2, v_2)) \in \gamma_{\mathbb{R}}\times\gamma_{\mathbb{R}} \mid \ell_1 = \ell_2\}$$
equipped with an almost complex structure $J$ given by $J((\ell, v_1), (\ell, v_2)) = ((\ell, -v_2), (\ell, v_1))$.
Given $((\ell, v_1), (\ell, v_2)) \in \gamma_{\mathbb{R}}\oplus\gamma_{\mathbb{R}}$, consider $(\ell, (\operatorname{span}_{\mathbb{C}}\{v_1\otimes 1 + v_2\otimes i\}, v_1\otimes 1 + v_2\otimes i))$. Note that $v_1 \in \ell$, so $\ell = \operatorname{span}_{\mathbb{R}}\{v_1\}$ and hence $i(\ell) = \operatorname{span}_{\mathbb{C}}\{v_1\otimes 1\}$. As $v_2 \in \ell = \operatorname{span}_{\mathbb{R}}\{v_1\}$, we have $v_2 = cv_1$ for some $c \in \mathbb{R}$ and hence \begin{align*} \operatorname{span}_{\mathbb{C}}\{v_1\otimes 1 + v_2\otimes i\} &= \operatorname{span}_{\mathbb{C}}\{v_1\otimes 1 + c(v_1\otimes i)\}\\ &= \operatorname{span}_{\mathbb{C}}\{(1 + ci)\cdot(v_1\otimes 1)\}\\ &= \operatorname{span}_{\mathbb{C}}\{v_1\otimes 1\}\\ &= i(\ell). \end{align*}
Therefore $(\ell, (\operatorname{span}_{\mathbb{C}}\{v_1\otimes 1 + v_2\otimes i\}, v_1\otimes 1 + v_2\otimes i)) \in i^*\gamma_{\mathbb{C}}$. So there is a real bundle map
\begin{align*} \Psi : \gamma_{\mathbb{R}}\oplus\gamma_{\mathbb{R}} &\to i^*\gamma_{\mathbb{C}}\\ ((\ell, v_1), (\ell, v_2)) &\mapsto (\ell, (\operatorname{span}_{\mathbb{C}}\{v_1\otimes 1 + v_2\otimes i\}, v_1\otimes 1 + v_2\otimes i)). \end{align*}
As $\Psi$ is injective and both bundles have real rank two, it is an isomorphism of real bundles. To see that $\Psi$ is actually an isomorphism of complex vector bundles, we have to check that it satisfies $\Psi\circ J = J'\circ \Psi$ where $J'$ is the almost complex structure on $i^*\gamma_{\mathbb{C}}$ which is given by $J'(\ell, (L, w)) = (\ell, (L, i\cdot w))$. Since
\begin{align*} \Psi(J((\ell, v_1), (\ell, v_2))) &= \Psi((\ell, -v_2), (\ell, v_1))\\ &= (\ell, (\operatorname{span}_{\mathbb{C}}\{-v_2\otimes 1 + v_1\otimes i\}, -v_2\otimes 1 + v_1\otimes i))\\ &= (\ell, (\operatorname{span}_{\mathbb{C}}\{i\cdot(v_1\otimes 1 + v_2\otimes i)\}, i\cdot(v_1\otimes 1 + v_2\otimes i)))\\ &= (\ell, (\operatorname{span}_{\mathbb{C}}\{v_1\otimes 1 + v_2\otimes i\}, i\cdot(v_1\otimes 1 + v_2\otimes i)))\\ &= J'(\Psi((\ell, v_1), (\ell, v_2))), \end{align*}
we finally conclude that $i^*\gamma_{\mathbb{C}} \cong \gamma_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}$.