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In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows:

  1. 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 $U$.

  2. 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?

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Given a connected cyclic cover $U'\to U$, you can normalize $X$ (a given compactification with snc boundary) in the fraction field of $U'$, yielding $p:X'\to X$. The multiplicities $\alpha_i$ come from ramification indices of $p$ along $D_i$. I guess you can find the line bundle $L$ in the $\mathbb{Z}/n$-decomposition of $p_* \mathcal{O}_{X'}$. –  Piotr Achinger Mar 4 at 20:08

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