Proposition 19.3.17.8D2910CD1CEC Let $\pi \colon Z\rightarrow X$, $p\colon Z'\rightarrow Z$ be proper bimeromorphic morphisms with $Z$ and $Z'$ being Kähler manifolds. Assume that $Y_{\bullet }$ admits a lifting $W_{\bullet }$ (resp. $W_{\bullet }'$) to $Z$ (resp. $Z'$). Then
1C158885A628
19.3.18
\begin{equation} \operatorname{cor}(Y_{\bullet },\pi \circ p)=\operatorname{cor}(Y_{\bullet },\pi )\operatorname{cor}(W_{\bullet },p). \end{equation}
Proof.
We let $\pi '=\pi \circ p$. Then $W'_{\bullet }$ is also the lifting of $W_{\bullet }$ with respect to $p$.
We argue by induction on $n\geq 1$. The case $n=1$ is trivial. Assume that $n\geq 2$ and the case $n-1$ has been solved. Then by (19.3.13), the desired formula (19.3.18) can be reformulated as
\[ \begin{aligned} & \begin{bmatrix} 1
& \nu _{W_1'\supseteq \cdots \supseteq W_ n'}((\pi '^*[Y_1]-[W_1'])|_{W_1'})
\\ 0
& \operatorname{cor}(Y_1\supseteq \cdots \supseteq Y_ n,\pi '|_{W_1'}:W_1'\rightarrow Y_1)
\end{bmatrix}=\\ & \begin{bmatrix} 1
& \nu _{W_1\supseteq \cdots \supseteq W_ n}((\pi ^*[Y_1]-[W_1])|_{W_1})
\\ 0
& \operatorname{cor}(Y_1\supseteq \cdots \supseteq Y_ n,\pi |_{W_1}:W_1\rightarrow Y_1)
\end{bmatrix}\cdot \\ & \begin{bmatrix} 1
& \nu _{W_1'\supseteq \cdots \supseteq W_ n'}((p^*[W_1]-[W_1'])|_{W_1'})
\\ 0
& \operatorname{cor}(W_1\supseteq \cdots \supseteq W_ n,p|_{W_1'}:W_1'\rightarrow W_1)
\end{bmatrix}\end{aligned}. \]
By the inductive hypothesis, this is equivalent to
\[ \begin{split} \nu _{W_1'\supseteq \cdots \supseteq W_ n'}\left((\pi '^*[Y_1]-[W_1'])|_{W_1'}\right)= \nu _{W_1'\supseteq \cdots \supseteq W_ n'}\left((p^*[W_1]-[W_1'])|_{W_1'}\right)+\\ \nu _{W_1\supseteq \cdots \supseteq W_ n}\left((\pi ^*[Y_1]-[W_1])|_{W_1}\right)\operatorname{cor}\left(W_1\supseteq \cdots \supseteq W_ n,p|_{W_1'}:W_1'\rightarrow W_1\right), \end{split} \]
which, thanks to proposition 19.3.14, can be further rewritten as
\[ \begin{split} \nu _{W_1'\supseteq \cdots \supseteq W_ n'}\left((\pi '^*[Y_1]-[W_1'])|_{W_1'}\right)= \nu _{W_1'\supseteq \cdots \supseteq W_ n'}\left((p^*[W_1]-[W_1'])|_{W_1'}\right)+\\ \nu _{W_1'\supseteq \cdots \supseteq W_ n'}\left(p|_{W_1'}^*\left((\pi ^*[Y_1]-[W_1])|_{W_1} \right)\right). \end{split} \]
Indeed, as divisors on $W_1'$,
\[ (\pi '^*[Y_1]-[W_1'])|_{W_1'}=(p^*[W_1]-[W_1'])|_{W_1'}+p|_{W_1'}^*((\pi ^*[Y_1]-[W_1])|_{W_1}). \]
The desired equality is therefore exactly the additivity of the valuation, proposition 19.3.9.
$\square$
Pome
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