relative canonical divisor VS relative differential sheaf

Question

Let $X$ be a normal variety, $f\colon X\rightarrow C$ a flat surjective morphism onto a smooth curve $C$ with connected fibers. After replacing $C$ by a finite covering, we may assume that $f$ has reduced fibers. Assume that there exists a Cartier divisor $D$ on $X$ with a nonzero morphism $\Omega_{X/C}^r\rightarrow \mathcal{O}(D)$, where $r$ is the relative dimension of $f$ and $\Omega^r_{X/C}$ is the r-th exterior power of the relative differential sheaf $\Omega_{X/C}$. My question is that if we can find an effective Weil $\Delta$ on $X$ such that $K_{X/C}+\Delta=D$. Of course, it is true if $f$ is smooth, and it also holds if $f$ is semi-stable.

I think this may be induced by the semi-stable reduction theorem, but I don't know how to lift our morphism of sheaves to some blow-up of $X$. Any comments or useful references are welcome.

Answer 1

This is true and actually has nothing to do with the morphism. It's a simple fact about divisors and their associated reflexive sheaves.

So, $\omega_{X/C}$, the reflexive sheaf of rank $1$ associated to the Weil divisor $K_{X/C}$ is the reflexive hull of $\Omega_{X/C}^r$. In particular, there exists a natural morphism $$ \Omega_{X/C}^r \to \omega_{X/C}, $$ which is neither necessarily injective nor surjective, but can be decomposed as a surjective morphism followed by an injective one. $$ \Omega_{X/C}^r \twoheadrightarrow (\Omega_{X/C}^r)/({\rm torsion}) \hookrightarrow \omega_{X/C}, $$

It is easy to see that if you have a morphism $$ \Omega_{X/C}^r \to \mathscr O(D), $$ to a torsion-free sheaf, then it factors through $(\Omega_{X/C}^r)/({\rm torsion})$. On the other hand, since $\omega_{X/C}$ is the reflexive hull of $\Omega_{X/C}^r$ and hence also of $(\Omega_{X/C}^r)/({\rm torsion})$, any reflexive sheaf (e.g., a line bundle) that contains $(\Omega_{X/C}^r)/({\rm torsion})$, also contains $\omega_{X/C}$, which translated to divisors means that $K_{X/C}\leq D$.

This means that what you would like is actually true. In fact, $D$ doesn't even have to be Cartier, it works if it is Weil divisor.

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Appendix (in response to the request in the comments):

As I mentioned, the morphism $\Omega_{X/C}^r \to \omega_{X/C}$ is simply the double dual. For any sheaf $\mathscr F$ you have a natural map $\mathscr F\to \mathscr F^{\vee\vee}$. This is it.

Another way to think about it is to consider the embedding of the open set $\imath: U=X\setminus {\rm Sing}\, X\hookrightarrow X$ and then $\mathscr F=\imath_*\imath^*\mathscr F$ so the above map is the natural map given by $1\to \imath_*\imath^*$.

For more on these issues, look at these MO answers: S2 property and canonical sheaf.