5.3. Log canonical divisors [016K]

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5.3. Log canonical divisors

If ${\mathcal{X}}$ is a regular model, $\pi\colon{\mathcal{X}}\to S$ is a locally complete intersection morphism, so the dualizing sheaf $\omega_{{\mathcal{X}}/S}$ is a well-defined line bundle (see [MN15, §4.1] for a more detailed discussion). For an arbitrary (normal) model, we may thus introduce the relative canonical divisor (class) $K_{{\mathcal{X}}/S}$ as the Weil divisor class on ${\mathcal{X}}$ such that ${\mathcal{O}}_{{\mathcal{X}}_{\mathrm{reg}}}(K_{{\mathcal{X}}/S})=\omega_{{\mathcal{X}}_{\mathrm{reg}}/S}$ . We then define:

(i)

the canonical divisor $K_{\mathcal{X}}:=K_{{\mathcal{X}}/S}+\pi^{*}K_{S}$ ;
(ii)

the log canonical divisor $K_{\mathcal{X}}^{\mathrm{log}}:=K_{\mathcal{X}}+{\mathcal{X}}_{0,\mathrm{red}}$ ;

(iii)

the relative log canonical divisor

K^{\mathrm{log}}_{{\mathcal{X}}/S}:=K^{\mathrm{log}}_{\mathcal{X}}-\pi^{*}K^{\mathrm{log}}_{S}=K_{{\mathcal{X}}/S}+{\mathcal{X}}_{0,\mathrm{red}}-{\mathcal{X}}_{0}.

Note that $K_{\mathcal{X}}^{\mathrm{log}}$ is ${\mathbb{Q}}$ -Cartier if and only if $K_{{\mathcal{X}}/S}^{\mathrm{log}}$ is ${\mathbb{Q}}$ -Cartier.

Example 5.1.

Assume that ${\mathcal{X}}$ is snc, and write as above ${\mathcal{X}}_{0}=\sum_{i\in I}b_{i}E_{i}$ . Pick a closed point $\xi\in{\mathcal{X}}_{0}$ , and denote by $J=\{0,\dots,p\}\subset I$ the set of components of ${\mathcal{X}}_{0}$ passing through $\xi$ . We may choose a regular system of parameters $z_{0},\dots,z_{n}\in{\mathcal{O}}_{{\mathcal{X}},\xi}$ such that $z_{i}$ is a local equation of $E_{i}$ for $0\leq i\leq p$ , i.e. ${t}=uz_{0}^{b_{0}}\dots z_{p}^{b_{p}}$ for some unit $u\in{\mathcal{O}}^{*}_{{\mathcal{X}},\xi}$ . The logarithmic form

\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}

is then a local generator of $K^{\mathrm{log}}_{{\mathcal{X}}}$ , and induces a local generator

\Omega^{\mathrm{rel}}:=\Omega\otimes(\frac{d{t}}{{t}})^{-1}

of $K^{\mathrm{log}}_{{\mathcal{X}}/S}$ .

Remark 5.2.

When ${\mathcal{X}}$ is snc, ${\mathcal{O}}_{\mathcal{X}}(K^{\mathrm{log}}_{{\mathcal{X}}/S})$ coincides with the relative logarithmic dualizing sheaf $\omega_{{\mathcal{X}}^{+}/S^{+}}$ of [NX13, (3.2.2)]. When ${\mathcal{X}}$ is regular, ${\mathcal{O}}_{\mathcal{X}}(K_{\mathcal{X}})$ is described in [dFEM11, Appendix A] as the determinant of the locally free sheaf $\Omega^{\prime}_{{\mathcal{X}}/k}\subset\Omega^{1}_{{\mathcal{X}}/k}$ of special differentials, corresponding to derivations $D$ of ${\mathcal{O}}_{\mathcal{X}}$ such that $D(f)=f^{\prime}({t})d{t}$ for $f\in k[\![{t}]\!]$ .