Let $n$ be a nonnegative integer, let $\mathbb{CP}^n$ denote complex projective $n$-space, let $L := \mathcal{O}_{\mathbb{CP}^n}(-1) \subset \mathbb{CP}^n \times \mathbb{C}^{n+1}$ be the tautological holomorphic line bundle with fiber $L_{[\ell]} = \ell$, and let $\mathcal{O}_{\mathbb{CP}^n}(1) := L^*$. Let $(e_0,\dots,e_n)$ be the standard basis of $\mathbb{C}^{n+1}$, and for each $0 \le i \le n$ let $Z_i \in H^0(\mathbb{CP}^n,\mathcal{O}_{\mathbb{CP}^n}(1))$ be the holomorphic section whose value at $[\ell]$ is the restriction to $\ell$ of the $i$-th coordinate functional on $\mathbb{C}^{n+1}$. Define $\alpha: \mathcal{O}_{\mathbb{CP}^n} \to \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus(n+1)}$ by $\alpha(1) = (Z_0,\dots,Z_n)$. Define $\beta: \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus(n+1)} \to T\mathbb{CP}^n$ fiberwise as follows: under the identification $\mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus(n+1)}_{[\ell]} \cong \ell^* \otimes \mathbb{C}^{n+1}$, an element $\lambda \otimes v$ maps to the tangent vector corresponding to the complex-[linear map](/page/Linear%20Map) $u \mapsto \lambda(u)(v \bmod \ell)$ from $\ell$ to $\mathbb{C}^{n+1}/\ell$ under the standard holomorphic identification $T_{[\ell]}\mathbb{CP}^n \cong \operatorname{Hom}_{\mathbb{C}}(\ell,\mathbb{C}^{n+1}/\ell)$. Then there is a short exact sequence of holomorphic vector bundles on $\mathbb{CP}^n$:

\begin{align*} 0 \longrightarrow \mathcal{O}_{\mathbb{CP}^n} \xrightarrow{\alpha} \mathcal{O}_{\mathbb{CP}^n}(1)^{\oplus(n+1)} \xrightarrow{\beta} T\mathbb{CP}^n \longrightarrow 0. \end{align*}