In the paper "A generalization of Kodaira-Ramanujam's vanishing theorem", Kawamata states a covering lemma (Lemma 5) which is



> 
> Let $X$ be a non-singular projective variety, and $D$ be a divisor with normal crossing on $X$. Let $m\_i$ be any positive integers attached to the irreducible components $D\_i$ of $D$. Then there are non-singular projective variety $X'$ and a finite flat morphism $f: X' \to X$ such that: (1) $f^\*D\_i= m\_iD'\_i$ for reduced effective divisors $D\_i'$; (2) the sum $\sum D\_i'$ is a divisor with normal crossing on $X'$.
> 
> 
> 


I am only interested in a special version of this result but I need more information.


Suppose $D$ itself is an ample divisor on $X$. According to the lemma, for any $m$, one has a finite morphism $f: X' \to X$ such that $f^\*D' = mD$ for some divisor $D'$ on $X'$.


I was wondering if the following are true:


(1) the degree of $f$ is a constant which only depends on $m$, and not depends on $\dim X$.


(2) one has $K\_{X'}=f^\*K\_{X}$.


I feel (2) is true because $\Omega\_{A/B}=0$ when $A$ is a finite $B$-module. However, according to the book "Introduction to the Mori Program" page 261, Lemma 5-2-4. One has


$$K\_{X'} = f^\*(K\_X + \frac{m-1}{m} D + \sum \frac{m-1}{m} G\_i)$$ where $\sum G\_i$ is some divisor, sharing no common component with $D$. I don't know where those $G\_i$ come from (sorry, but I was unable to follow the proof given there). It is those $G\_i$ that give me real trouble. Hence, I was wondering even if $K\_{X'}=f^\*K\_{X}$ is not valid, can we get rid of those $G\_i$? Or at least has some control on them in the sense their top intersection number is bounded?



A covering lemma of Kawamata
I'm afraid that what you would like is not going to happen.


The problem with achieving (1) is that the procedure requires a step when you take a multiple of an ample line bundle that is very ample and then that multiple will contribute to the degree of the map. So, this degree will depend on $X$. I suppose you may be able to get something better if $K\_X$ is ample, but then you might run into a problem if the number of components of $D$ is not bounded. The procedure makes a cover for each irreducible component separately. If your $D$ is irreducible and you don't mind making your $m$ depend on the multiple you have to take to get to very ampleness, then you might have a chance.


On the other hand as they say a snowflake has a better chance to go to hell and come back, then to force this cover to be crepant. Your mistake is that it is not true that $\Omega\_{A/B}=0$ unless $A$ is smooth (a.k.a. étale) over $B$. You might even have an $X$ that does not admit a non-trivial étale cover. You can see easily that $\mathbb P^1$ is such: If $f:Y\to \mathbb P^1$ is a finite morphism of degree $d$ from a non-singular projective variety $Y$, then $Y$ is a curve as well. If $f^\*K\_{\mathbb P^1}=K\_Y$, then $\deg K\_Y=-2d$. Let's say $Y$ is of genus $g$, then we get $-2d=2g-2$. Since both $d$ and $g$ are non-negative integers, this can only happen if $d=1$ and $g=0$ and $f$ is an isomorphism.


What you are missing is the **Hurwitz formula**. You can look that up in any basic algebraic geometry book. 


