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Corollary 18.1.39F7A7811A159E

Corollary 18.1.39.F7A7811A159E Let $\pi \colon X\rightarrow Z$ be a proper bimeromorphic morphism from $X$ to a Kähler manifold $Z$. Then

4A8B3C0B6CE6 18.1.40
\begin{equation} \operatorname{vol}(T_1,\ldots ,T_ n)=\operatorname{vol}(\pi _*T_1,\ldots ,\pi _*T_ n). \end{equation}

Proof. Observe that we may assume that $T_ i=\operatorname{Reg}T_ i$ for all $i=1,\ldots ,n$. In fact, since the pushforward of the divisorial part of $T_ i$ is divisorial as well, hence by proposition 18.1.27 5, they do not contribute to the volumes.

Now by proposition 18.1.37, it remains to show that

\[ \operatorname{vol}(T_1,\ldots ,T_ n)=\operatorname{vol}(\pi ^*\pi _*T_1,\ldots ,\pi ^*\pi _*T_ n). \]

By 18.1.38, the difference $\pi ^*\pi _*T_ i-T_ i$ is divisorial, hence our desired equality follows from proposition 18.1.27 5. $\square$

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