I try to prove that $\operatorname{Pic}^0(\mathbb{P}^n)=0$. We have the exponential exact sequence
$$0 \to \mathbb{Z} \to \mathbb{C} \to \mathbb{C}^* \to 0$$
and the exponential exact sequence of sheaves on any complex manifold $M$:
$$0 \to \mathbb{Z} \to \mathcal{O}_M \to \mathcal{O}^*_M \to 0.$$
Then, if $M$ is compact we have the exact sequence
$$0 \to \mathbb{Z}(M) \to \mathcal{O}_M (M) \to \mathcal{O}^*_M(M) \to 0,$$
because we have $\mathbb{Z}(M)=\mathbb{Z}$ by connectedness, and $M$ is a compact complex manifolds gives us $\mathcal{O}_M (M)=\mathbb{C}$, $\mathcal{O}^*_M(M)=\mathbb{C}^*$. We have the sequence of Cech cohomology:
$$0 \to \mathbb{Z}(M) \to \mathcal{O}_M (M) \to \mathcal{O}^*_M(M) \to H^1(M,\mathbb{Z}) \to^j H^1(M,\mathcal{O}_M) \to^k H^1(M,\mathcal{O}^*_M) \to^c H^2(M, \mathbb{Z}).$$
The set $\operatorname{Pic}^0(M) := H^1(M,\mathcal{O}_M)/Im(H^1(M,\mathbb{Z}))$ by definition. The $j$ in the exact sequence is always injective.
If I take $M=\mathbb{P}^n$, why $j$ is 1-1 maps? It suffice to prove $k=0$ or $c$ is injective. Or, we can prove that $H^1(\mathbb{P}^n, \mathcal{O})=0$. but I'm not sure that's true for $n \geq 2$, for $n=1$ it's true.
Any hint to prove that?