A covering lemma of Kawamata

Question

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?

Answer 1

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.