# Non-constant holomorphic map onto a smooth curve

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!

Let me show that such a map usually doesn't exist (even if we don't remove any additional curves). Consider the case when $$\Gamma$$ is a curve of degree $$\ge 4$$. Then one can slightly perturb $$\Gamma$$ in $$\mathbb CP^2$$ to a curve $$\Gamma'$$ that is not isomorphic to $$\Gamma$$ (so that it stays in a small neighbourhood of $$\Gamma$$). If the map $$\pi$$ would exists, we can restrict it to $$\Gamma'$$, and it will be extendable to the whole $$\Gamma'$$. So we get a holomorphic (non constant) map $$\pi: \Gamma'\to \Gamma$$, which is impossible, since $$\Gamma$$ and $$\Gamma'$$ are not isomorphic (and by Riemann-Hurwitz for two curves of the same genus $$\ge 2$$ a non-constant holomorphic map $$\Gamma\to \Gamma'$$ exits only if the curves are isomorphic).
This argument can be easily generalised to curves $$\Gamma$$ of degree $$3$$, i.e. cubics. The case of conics and lines should be analysed separately (and for lines $$\Gamma$$ such a map sometimes exists, for example, when all $$\Gamma_i$$'s are lines transverse to the line $$\Gamma$$ and intersecting at one point).