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Suppose $f: X \to Y$ is a morphism between varieties, let $y \in Y$ be a closed point, and $X_y$ be the fibre of the morphism over $y$. I learned that the canonical divisor $K_X$ of $X$, the canonical divisor $K_{X_y}$ of $X_y$, and the relative canonical divisor $K_{X/Y}:= K_X -f^*K_Y$ are somehow related as the following:

(1) For generic $y \in Y$, one has $ K_{X_y} \cong K_X|_{X_y}$.

Remark Intuitively, I can understanding the relation given above, but I do not know how to show this rigorously. Besides, I do not know the correct conditions to be put on $ f, X, Y$. Certainly one should assume $X,Y$ at least to be normal, but it seems that it is too strong to assume they are smooth . However, you can freely assume they are projective/proper varieties.

(2) For generic $y$, $K_{X_y} \cong K_{X/Y}|_{X_y}$.

Remark (i) $Y$ should to be at least $\mathbb{Q}$-Gorenstein in order to make sense of pull-back of $K_Y$. (ii) If (1) is true, then (2) is equivalent to $(f^*K_Y)|_{X_y}=0$, this seems like some projection formula. (3) Both formula are not valid for all $y \in Y$, this can be seen from the example of $f$ being the blowup of one point in $\mathbb{P}^2$.

Li Yutong
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  • The basic point to understand is: what is the normal bundle of a smooth fibre of $f$? (Assume $X$ is smooth, so by Bertini the general fibre of $f$ is smooth.) Once you know that, the "adjunction sequence" you can find in Hartshorne gives the answer. –  Sep 03 '13 at 14:51
  • So also the paper https://msp.org/ant/2013/7-2/ant-v7-n2-p04-s.pdf . Which is called Kollár condition. It's not true without control of singularity. – yi li Feb 01 '24 at 03:08

1 Answers1

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Given a birational morphism $f:Y−→X$ with $Y$ normal, the relative canonical divisor, denoted by $K_{Y/X}$, is the unique $\mathbb Q$-divisor on $Y$ which is supported on the exceptional locus and linearly equivalent to $K_Y-f^∗(K_X)$. Here $f^∗(K_X)$ is defined as $\frac{1}{r}f^∗(rK_X)$ where $r$ is such that it becomes $\mathbb Q$-Gorenstein. If both $X$ and $Y$ are smooth varieties, $K_{Y/X}$ is an effective divisor locally defined by the Jacobian determinant of $f$ .

By the adjunction formula we have $$K_{X_y}\cong K_X |_{X_y} ⊗ \det(N_{{X_y}/X}),$$

where $N_{X_y}/X$ is the normal bundle of $X_y$ inside $X$. However this normal bundle is trivial, because its dual is globally trivialized by $f^∗(dy_1 ∧ · · · ∧ dy_m),$ where $(y_1, . . . , y_m)$ are local holomorphic coordinates on $Y$ near $y$ . See Lemma 5.6 for more details .