$\overline{C}_{n, n-1}$ Revisited

Abstract

In this chapter, we give a detailed proof of the subadditivity theorem of the logarithmic Kodaira dimension for morphisms of relative dimension one.

5.4 Appendix: A Vanishing Lemma

In this appendix, we see that we can quickly recover [Kaw2, Lemma 4] by the Kawamata–Viehweg vanishing theorem.

Lemma 5.4.1

([Kaw2, Lemma 4]) Let $f:X_1 \rightarrow X_2$ be a birational morphism of smooth complete varieties and let $D_1$ and $D_2$ be simple normal crossing divisors on $X_1$ and $X_2$, respectively. We assume that $D_1={{\text {Supp}}}f^*D_2$. Then we have

$$ Rf_*\mathcal O_{X_1}(-D_1)\simeq \mathcal O_{X_2}(-D_2). $$

Proof

Note that $D_1=\lceil \varepsilon f^*D_2\rceil$ for $0<\varepsilon \ll 1$. Since f is birational, $\varepsilon f^*D_2$ is f-nef and f-big. Therefore, by the relative Kawamata–Viehweg vanishing theorem, we have $R^if_*\mathcal O_{X_1}(K_{X_1}+D_1)=0$ for every $i>0$ (see, for example, [F12, Theorem 3.2.1]). We write

$$ K_{X_1}=f^*(K_{X_2}+D_2)+\sum _E a(E, X_2, D_2)E. $$

Since $K_{X_2}+D_2$ is Cartier, $a(E, X_2, D_2)\in \mathbb Z$ for every E. Since $X_2$ is smooth and $D_2$ is a simple normal crossing divisor on $X_2$, it is well known that $a(E, X_2, D_2)\ge -1$ for every E, that is, $(X_2, D_2)$ is lc (see, for example, [F12, Lemma 2.3.9]). We can easily see that $f(E)\subset {{\text {Supp}}}D_2$ if $a(E, X_2, D_2)=-1$. Therefore, we obtain

$$ K_{X_1}+D_1=f^*(K_{X_2}+D_2)+F $$

for some effective f-exceptional Cartier divisor F on $X_1$. Thus, we obtain $f_*\mathcal O_{X_1}(K_{X_1}+D_1)\simeq \mathcal O_{X_2}(K_{X_2}+D_2)$. This means that

$$ Rf_*\mathcal O_{X_1}(K_{X_1}+D_1)\simeq \mathcal O_{X_2}(K_{X_2}+D_2). $$

By Grothendieck duality, we have

$$\begin{aligned} Rf_*\mathcal O_{X_1}(-D_1)&\simeq R\mathcal H om (Rf_*\mathcal O_{X_1} (K_{X_1}+D_1), \mathcal O_{X_2}(K_{X_2})) \\&\simeq \mathcal H om (\mathcal O_{X_2}(K_{X_2}+D_2), \mathcal O_{X_2}(K_{X_2})) \\&\simeq \mathcal O_{X_2}(-D_2). \end{aligned}$$

This is the desired quasi-isomorphism.    $\square$

Note that the Kawamata–Viehweg vanishing theorem was not known when Kawamata wrote [Kaw2]. The reader can find various formulations and some generalizations of the Kawamata–Viehweg vanishing theorem in [F12]. We also note that we did not use [Kaw2, Lemma 4] in the proof of Theorem 5.2.3 in Sect. 5.2.