Let $\Gamma$ be a smooth projective curve in $\mathbb{P}^2$ and let $U$ be an open neighborhood of $\Gamma$. Denote by $\Gamma_1,\Gamma_2,\ldots,\Gamma_n$ a finite collection of smooth curves intersecting $\Gamma$ transversally. My question is:

**Does there exist a non-constant holomorphic function $\pi: U \setminus \bigcup\limits_{i=1}^n \Gamma_i \to \Gamma \setminus \bigcup\limits_{i=1}^n \Gamma_i$** ?

The situation as in the following picture.

I can build a smooth map $\pi$ easily by partition of unity. Intuitively, $\pi$ seems like the tubular neighborhood projection of $\Gamma$ ? But unfortunately, a holomorphic tubular neighborhood does not exist in general as discussed in Is there any holomorphic version of the tubular neighborhood theorem?. And I realized that even the existence of a non-constant holomorphic map $\pi : U \to \Gamma$ (without removing any curve) is not obvious to me.

Any comment is welcome. Thanks in advance!