# Relative canonical divisors

Suppose that $X$ is a Gorenstein variety and that $\pi : Y \to X$ is a birational map of varieties with normal $Y$.

In this case the relative canonical divisor is defined to be $K_Y - \pi^*K_X$ (if you choose $K_Y$ and $K_X$ that agree where $\pi$ is an isomorphism, $K_Y - \pi^*K_X$ is independent of the particular representatives of $K_X$ and $K_Y$ you pick). $\mathcal{O}_Y(K_Y - \pi^* K_X)$ can also be identified with $\pi^! \mathcal{O}_X$ in this case.

Suppose that that $X$ is $\mathbb{Q}$-Gorenstein (which means that $nK_X$ is Cartier for some $n > 0$, also assume Cohen-Macaulay if it helps). In birational geometry, the relative canonical divisor (a $\mathbb{Q}$-divisor) is defined by the formula $$K_{Y/X} = K_Y - {1 \over n} \pi^* (nK_X).$$ Again, if you choose $K_X$ and $K_Y$ that agree where $\pi$ is an isomorphism, then this $\mathbb{Q}$-divisor $K_{Y/X}$ is independent of the choices involved.

Question: Is it known whether $\mathcal{O}_Y(\lceil K_Y - {1 \over n} \pi^* (nK_X) \rceil)$ (or maybe with $\lfloor \cdot \rfloor$) is at all related to the zeroth cohomology of $\pi^! \mathcal{O}_X$?

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