Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \to \mathbb{P}^2$ branched over $\Lambda$ defined by the function field,
$$ \mathbb{C}(\mathbb{P}^2)( (\ell_1/\ell_k)^{1/n}, \cdots, (\ell_{k-1}/\ell_k)^{1/n}) $$ where $\ell_1, \dots, \ell_k$ are linear forms defining the $k$ lines of $\Lambda$.
I am wondering if the fundamental group of $H(\Lambda, n)$ has been studied in terms of the arrangement $\Lambda$. In particular, have the $\Lambda$ such that $\pi_1(H(\Lambda, n)) = 0$ (or finite is also interesting) been classified?
I know that both K3 surfaces and $K(\pi, 1)$ general type surfaces appear as examples so the answer to this question may be quite complicated.
Also, does anyone know a reference that computes the Hodge numbers of $H(\Lambda, n)$?
