In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows:
Let $X$ be a smooth projective variety over some field $k$ of characteristic zero and let $D=\sum \alpha_i D_i$ be an effective (but not necessarily reduced) normal crossings divisor on $X$. Assume that there exists a line bundle $L$ and an integer $d$ such that $L^d=\mathcal{O}_X(D)$. Then these data gives a normal cyclic covering $\pi: Y \to X$ which is unramified at $U$.
Suppose now you start with $U$ and with a representation $\rho: \pi_1(U, \overline{u}) \to GL(n, \mathbb{C})$ of the algebraic fundamental group of $U$ with some geometric base point $\overline{u}$. Assume that the image of $\rho$ is a finite cyclic group. Then to $\rho$ should correspond a finite cyclic covering of $U$.
My question is how 1 and 2 come together. Is the finite cyclic covering given by 2 of the form in 1. Of course if you want to get the normal crossings divisor you can just compactify $U$. But how are you going to get the multiplicities of the components?
Another question: is there a similar description for finite not necessarily cyclic covers?
