Let $X$ be a connected complex manifold such that $X$ has a complex submanifold $Y$ so that $f:Y\to \mathbb C$ is holomorphic iff $f$ is constant. Now if $Y$ is a closed submanifold, then $X$ may have non-constant holomorphic function e.g. $g:T\times \mathbb C\to \mathbb C;\ (t,z)\mapsto z$. Where $T$ is a complex torus.
But what if $Y$ is not closed, can $X$ have non-constant holomorphic functions?
Best Answer
Yes, it is possible. Suppose $Z$ is a compact complex manifold of complex dimension at least $2$. Let $Y$ be $Z$ minus a point (or any other compact subset $K$ so that $Z\setminus K$ is connected) - then by Hartog's Extension Theorem, the holomorphic functions on $Y$ and $Z$ are exactly the same. So $Y\times\{0\}\subset Z\times\Bbb C$, and by the method of the example in your post, one finds non-constant holomorphic functions.