Proving that a holomorphic function is constant

Question

I am attempting to prove the following: Let $X$ be a connected complex manifold, and $f\in \mathcal{O}(X)$. For any $x\in X$, there is a complex submanifold of $X$ which is biholomorphic to $\mathbb{P}(\mathbb{C})$ and contains $x$. Does it follow that $f$ is constant?

Thanks in advance.

Answer 1

This is not true, if I understand the statement correctly. For instance, there are nonconstant holomorphic functions on $X = \mathbb C \times \mathbb P^1(\mathbb C)$, even though every $x \in X$ is contained in a closed submanifold of $X$ which is biholomorphic to $\mathbb P^1(\mathbb C)$.