Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.)

Is it true that for any open neighborhood $U$ (in the analytic topology) of $\mathbb{C}\mathbb{P}^1$ there exists a smaller neighborhood $V$ of the latter such that $$H^i(V, \mathcal{O})=0 \mbox{ for any } i>0,$$ where $\mathcal{O}$ is the structure sheaf?