the natural isomorphism $H^0(\mathbb P^n, T_{\mathbb P^n}(-1)) \cong H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(1))^\vee $

Question

I have trouble in proving the natural isomorphism $$H^0(\mathbb P^n, T_{\mathbb P^n}(-1)) \cong H^0(\mathbb P^n, \mathcal O_{\mathbb P^n}(1))^\vee $$ where '$\vee$' stands for the dual space, and $T_{\mathbb P_n}:= \Omega_{\mathbb P^n}^\vee$ is the holomorphic tangent bundle of $\mathbb P^n$.

My attempt is to use the Euler sequence $$ 0\to \Omega_{\mathbb P^n} \to \mathcal O_{\mathbb P^n}(-1)^{\oplus n+1} \to \mathcal O_{\mathbb P^n} \to 0 $$ Taking duals and tensoring by $\mathcal O_{\mathbb P^n}(-1) $ yields another short exact sequence: $$ 0\to \mathcal O_{\mathbb P^n}(-1) \to \mathcal O_{\mathbb P^n}^{\oplus n+1} \to T_{\mathbb P^n}(-1) \to 0 $$ I also think of Serre duality to establish the right side of the isomorphism. Anyway, I need help. Thanks!.

Answer 1

An idea : the last sequence you wrote give an isomorphism $H^0(P^n,\mathcal O^{\oplus n+1}) \cong H^0(P^n, T_{P^n}(-1))$ as $\mathcal O(-1)$ is acyclic (i.e $H^i(P^n, \mathcal O(-1)) = 0$ for all $i \in \Bbb N$). On the other hand, I believe that a section of $(t_0, \dots, t_n) \in H^0(P^n, \mathcal O^{\oplus (n+1)})$ can be interpreted as a linear functional on $H^0(P^n, O(1))$, since the global sections are the vector space generated by $x_0, \dots, x_n$.