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, holomorphic tubular neighborhood does not exist in general. Thank in advanced !