2 typo

Let $n>0$. It is enough to show that the vector bundles $\mathcal{O} \oplus \mathcal{O}(n)$ and $\mathcal{O}(1) \oplus \mathcal{O}(n-1)$ are isomorphic as complex vector bundles with smooth transition maps. (They are NOT isomorphic as complex vector bundles with holomorphic transition maps.) Once we show that, we get that $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(n))$ is diffeomorphic to $\mathbb{P}(\mathcal{O}(1) \oplus \mathcal{O}(n-1))$, and we have $\mathcal{O}(1) \oplus \mathcal{O}(n-1) \cong \mathcal{O}(1) \otimes ( \mathcal{O} \oplus \mathcal{O}(n-2))$, so $\mathbb{P}(\mathcal{O}(1) \oplus \mathcal{O}(n-1)) \cong \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(n-2))$.

We have a short exact sequence $$0 \to \mathcal{O} \to \mathcal{O}(1) \oplus \mathcal{O}(n-1) \to \mathcal{O}(n) \to 0$$ where the maps are given by $\left( \begin{smallmatrix} x \\ y^{n-1} \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} -y^{n-1} & x \end{smallmatrix} \right)$.

Now, in the category of smooth vector bundles, every short exact sequence $0 \to A \to B \to C \to 0$ splits. The proof is as follows: use a partition of unity argument to but put a positive definite Hermitian structure on $A$, $B$ and $C$. Then the adjoint of the map $B \to C$ provides a splitting.

So $\mathcal{O}(1) \oplus \mathcal{O}(n-1) \cong \mathcal{O} \oplus \mathcal{O}(n)$ and we are done.

1

Let $n>0$. It is enough to show that the vector bundles $\mathcal{O} \oplus \mathcal{O}(n)$ and $\mathcal{O}(1) \oplus \mathcal{O}(n-1)$ are isomorphic as complex vector bundles with smooth transition maps. (They are NOT isomorphic as complex vector bundles with holomorphic transition maps.) Once we show that, we get that $\mathbb{P}(\mathcal{O} \oplus \mathcal{O}(n))$ is diffeomorphic to $\mathbb{P}(\mathcal{O}(1) \oplus \mathcal{O}(n-1))$, and we have $\mathcal{O}(1) \oplus \mathcal{O}(n-1) \cong \mathcal{O}(1) \otimes ( \mathcal{O} \oplus \mathcal{O}(n-2))$, so $\mathbb{P}(\mathcal{O}(1) \oplus \mathcal{O}(n-1)) \cong \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(n-2))$.

We have a short exact sequence $$0 \to \mathcal{O} \to \mathcal{O}(1) \oplus \mathcal{O}(n-1) \to \mathcal{O}(n) \to 0$$ where the maps are given by $\left( \begin{smallmatrix} x \\ y^{n-1} \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} -y^{n-1} & x \end{smallmatrix} \right)$.

Now, in the category of smooth vector bundles, every short exact sequence $0 \to A \to B \to C \to 0$ splits. The proof is as follows: use a partition of unity argument to but a positive definite Hermitian structure on $A$, $B$ and $C$. Then the adjoint of the map $B \to C$ provides a splitting.

So $\mathcal{O}(1) \oplus \mathcal{O}(n-1) \cong \mathcal{O} \oplus \mathcal{O}(n)$ and we are done.