Pome

Proposition 18.1.3724049C5AD956

Proposition 18.1.37.24049C5AD956 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$ to $X$. Then

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

Proof. Using the definition of the mixed volume, we may easily reduce to the case where $\operatorname{vol}T_ i>0$ for each $i=1,\ldots ,n$. By proposition 16.6.10, we know that if we write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$, then

\[ \pi ^*P_{\theta _ i}[\varphi _ i]_{\mathcal{I}}=P_{\pi ^*\theta _ i}[\pi ^*\varphi _ i]_{\mathcal{I}} \]

for each $i=1,\ldots ,n$. In particular,

\[ \operatorname{vol}\pi ^*T_ i=\operatorname{vol}T_ i>0. \]

Our assertion follows from the bimeromorphic invariance of the non-pluripolar product proposition 13.13.4. $\square$

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