PomeDefinition 19.3.12.E494C67EB4B8 Let $\pi \colon Z\rightarrow X$ be a proper bimeromorphic morphism where $Z$ is a Kähler manifold. We say that a smooth flag $W_{\bullet }$ on $Z$ is a lifting of $Y_{\bullet }$ to $Z$ if $\pi (W_ i)\subseteq Y_ i$ and the induced map $W_ i\to Y_ i$ is bimeromorphic for each $i=0,\ldots ,n$.
In this case, we define $\operatorname{cor}(Y_{\bullet },\pi )\in \operatorname{Aut}(\mathbb {Z}^ n_{\mathrm{lex}})$ inductively as follows: When $n=1$, we define $\operatorname{cor}(Y_{\bullet },\pi )=[1]$; when $n>1$, we set
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19.3.13
\begin{equation} \operatorname{cor}(Y_{\bullet },\pi )\coloneqq \begin{bmatrix} 1
& \nu _{W_1\supseteq \cdots \supseteq W_ n}\Bigl ((\pi ^*[Y_1]-[W_1])|_{W_1}\Bigr )
\\ 0
& \operatorname{cor}\Bigl (Y_1\supseteq \cdots \supseteq Y_ n,\pi |_{W_1}\colon W_1\rightarrow Y_1\Bigr )
\end{bmatrix}. \end{equation}
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