Suppose $\mathbb{P}^1$ is smoothly embedded as a real submanifold in $\mathbb{P}^3$. Is $\mathbb{P}^1$ then necessarily a complex submanifold of $\mathbb{P}^3$?
If this is not true in general, what can obstruct $\mathbb{P}^1$ from being a complex submanifold?
Supposing $\mathbb{P}^1$ is in fact a complex submanifold, what is the first Chern class $c_1(N)$ of the normal bundle $N$ of $\mathbb{P}^1$? Grothendieck's theorem gives that $N$ is equivalent to $\mathcal{O}_{\mathbb{P}^1}(a_1)\oplus \mathcal{O}_{\mathbb{P}^1}(a_2)$ where $a_1\geq a_2$ and $a_1+a_2=c_1(N)$. Can the normal bundle (i.e. $a_1$ and $a_2$) be determined in this case?