Let $X,Y$ be complex manifolds of $\dim X=n$, $\dim Y=m>1$, $U\subset X$ open and $g\colon U\to Y$ holomorphic embedding. Then $g(U)$ is a submanifold of codimension $m-n\ge1$. It seems clear that $Y\setminus g(U)$ is connected, think for example at $\Bbb C\setminus\{0\}$. I'm searching for a reference, as it seems a well known fact.



Complement of complex submanifolds of codimension $\ge1$ is connected
Every (real) codimension $2^+$ embedding has path connected complement.


The proof is by using the tubular neighborhood theorem to reduce to showing that the total space of the normal bundle of $g(U)$ in $Y$ minus the zero section is connected, and noticing that the normal bundle is a rank $2^+$ vector bundle.


See [this MO answer](https://mathoverflow.net/questions/71895/on-a-special-case-of-alexander-duality).


