Tautological line bundle of a projective subspace

Question

Let $W$ be a complex vector space and $V\subset W$ a linear subspace. We get then an induced inclusion $$i: \mathbb{P}(V)\hookrightarrow \mathbb{P}(W).$$ Denote by $\mathcal{O}_V(-1)$ and $\mathcal{O}_W(-1)$ the tautological line bundles of $\mathbb{P}(V),\mathbb{P}(W)$. What relation is there between $\mathcal{O}_V(-1)$ and $\mathcal{O}_W(-1)$? Intuitively I would say that there is an isomorphism $$\mathcal{O}_V(-1)\simeq i^*\mathcal{O}_W(-1)$$ Is that true? Is there more than that?

Answer 1

Recall that $\mathcal{O}_W(-1) = \{(w, L) \in W\times\mathbb{P}(W) \mid w \in L\}$ and the map $\pi_W : \mathcal{O}_W(-1) \to \mathbb{P}(W)$ is given by $(w, L) \to L$. The inclusion map $V \hookrightarrow W$, which I will denote by $\iota$, induces an inclusion $i : \mathbb{P}(V) \hookrightarrow \mathbb{P}(W)$, and these inclusions preserves the incidence correspondence, i.e. if $v \in \ell$, then $\iota(v) \in i(\ell)$.

Consider the map $\mathcal{I} : \mathcal{O}_V(-1) \to \mathcal{O}_W(-1)$ given by $(v, \ell) \mapsto (\iota(v), i(\ell))$; this map is defined because the inclusions preserve the incidence correspondence. Note that $\mathcal{I}$ is linear on fibers and $\pi_W\circ\mathcal{I} = \mathcal{I}\circ\pi_V$, so $\mathcal{I}$ is a vector bundle homomorphism covering $i$.

$$\require{AMScd} \begin{CD} \mathcal{O}_V(-1) @>{\mathcal{I}}>> \mathcal{O}_W(-1)\\ @V{\pi_V}VV @VV{\pi_W}V \\ \mathbb{P}(V) @>{i}>> \mathbb{P}(W) \end{CD}$$

As $\iota$ is injective, it follows that $\mathcal{I}$ is an injective vector bundle homomorphism between rank one vector bundles, and hence $\mathcal{I}$ is an isomorphism covering $i$. Therefore $\mathcal{O}_V(-1) \cong i^*\mathcal{O}_W(-1)$.

If you would like an explicit isomorphism, note that

\begin{align*} i^*\mathcal{O}_W(-1) &= \{(\ell, (w, L)) \in \mathbb{P}(V)\times\mathcal{O}_W(-1) \mid i(\ell) = \pi_W(w, L)\}\\ &= \{(\ell, (w, L)) \in \mathbb{P}(V)\times\mathcal{O}_W(-1) \mid i(\ell) = L\} \end{align*}

with projection map $i^*\mathcal{O}_W(-1) \to \mathbb{P}(V)$ given by $(\ell, (w, i(\ell))) \mapsto \ell$. As above, one can show that the map $\mathcal{O}_V(-1) \to i^*\mathcal{O}_W(-1)$ given by $(v, \ell) \mapsto (\ell, (\iota(v), i(\ell)))$ is an isomorphism - note that this is essentially equivalent to $\mathcal{I}$.