We first extend the notion of volume in definition 16.6.13 to the mixed case. Let $\theta _1,\ldots ,\theta _ n$ be smooth closed real $(1,1)$-forms on $X$ representing pseudo-effective classes.
Definition 18.1.21.EDCDF35668C4 Let $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ for $i=1,\ldots ,n$. Write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for each $i=1,\ldots ,n$. Assume that $\operatorname{vol}T_ i>0$ for all $i=1,\ldots ,n$. We define the mixed volume $\operatorname{vol}(T_1,\ldots ,T_ n)$ as follows:
65AD0F03CD20
18.1.22
\begin{equation} \operatorname{vol}(T_1,\ldots ,T_ n)=\int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right). \end{equation}
When $\operatorname{vol}T_ i>0$ for each $i=1,\ldots ,n$, the definition (
18.1.22) does not depend on how we represent $T_ i$ as $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$, this is a consequence of theorem
13.13.11 and proposition
16.6.9.
Lemma 18.1.23.20316D4886CF
4642B46CE130
The mixed volume is symmetric in its $n$-variables.
2AE20B98C203
For any Kähler form $\omega $ on $X$, we have
7F052390F186
18.1.24
\begin{equation} \operatorname{vol}\left(T_1,\ldots ,T_ n\right)=\lim _{\epsilon \to 0+} \operatorname{vol}\left(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega \right). \end{equation}
D321C1B261ED
Consider closed positive $(1,1)$-currents $T_1,\ldots ,T_ n,T_1'$ on $X$ with positive volumes, then
7.10
\begin{equation} \operatorname{vol}(T_1+T_1',T_2,\ldots ,T_ n)=\operatorname{vol}(T_1,T_2,\ldots ,T_ n)+\operatorname{vol}(T_1',T_2,\ldots ,T_ n). \end{equation}
7787D4E19E46
Consider closed positive $(1,1)$-currents $T_1,\ldots ,T_ n$ on $X$ with positive volumes and $\lambda > 0$, then
\[ \operatorname{vol}(\lambda T_1,T_2,\ldots ,T_ n)=\lambda \operatorname{vol}(T_1,T_2,\ldots ,T_ n). \]
DAD3C89C5ACC
Suppose that $T_1,\ldots ,T_ n,S_1,\ldots ,S_ n$ are closed positive $(1,1)$-currents on $X$ with positive volumes such that $T_ i\preceq _{\mathcal{I}} S_ i$ and $\{ T_ i\} =\{ S_ i\} $ for each $i=1,\ldots ,n$. Then
7.11
\begin{equation} \operatorname{vol}(T_1,\ldots ,T_ n)\leq \operatorname{vol}(S_1,\ldots ,S_ n). \end{equation}
The notation $\operatorname{Reg}$ is defined in (
5.19.13).
Proof.
1 This follows from the symmetry of the mixed non-pluripolar product in (18.1.22), cf. proposition 13.11.4 2.
2 For each $i$ and each $\epsilon >0$, we have
\[ P_{\theta _ i}[\varphi _ i]_{\mathcal{I}}\sim _ P P_{\theta _ i+\epsilon \omega }\left[P_{\theta _ i}[\varphi _ i]_{\mathcal{I}}\right]_{\mathcal{I}}=P_{\theta _ i+\epsilon \omega }[\varphi _ i]_{\mathcal{I}} \]
as a consequence of example 18.1.11. Hence using proposition 16.8.22 and the multi-linearity of the non-pluripolar product proposition 13.11.4 6,
\[ \begin{aligned} & \lim _{\epsilon \to 0+}\operatorname{vol}(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega )\\ =& \lim _{\epsilon \to 0+}\int _ X \left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}\right)\wedge \cdots \wedge \left(\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ =& \operatorname{vol}(T_1,\ldots ,T_ n). \end{aligned} \]
3 We write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ as before for each $i$ and $T_1'=\theta _1'+\mathrm{dd}^{\mathrm{c}}\varphi _1'$. The potentials $P_{\theta _1}[\varphi _1]_{\mathcal I}$ and $P_{\theta _1'}[\varphi _1']_{\mathcal I}$ are $\mathcal I$-model with positive masses, hence $\mathcal I$-good. By proposition 18.1.17, their sum is $\mathcal I$-good. Moreover, it is $\mathcal I$-equivalent to $\varphi _1+\varphi _1'$. Then thanks to proposition 18.1.17,
\[ P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _1'}[\varphi _1']_{\mathcal{I}}\sim _ P P_{\theta _1+\theta _1'}\left[P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _1'}[\varphi _1']_{\mathcal{I}} \right]_{\mathcal{I}} = P_{\theta _1+\theta _1'}\left[\varphi _1+\varphi _1'\right]_{\mathcal{I}} \]
Thus by proposition 16.8.22, we have
\[ \begin{aligned} & \operatorname{vol}(T_1+T_1',T_2,\ldots ,T_ n)\\ =& \int _ X \left( \theta _1+\theta _1'+\mathrm{dd}^{\mathrm{c}}P_{\theta _1+\theta _1'}[\varphi _1+\varphi _1']_{\mathcal{I}} \right)\wedge \left( \theta _2+\mathrm{dd}^{\mathrm{c}}P_{\theta _2}[\varphi _2]_{\mathcal{I}}\right)\wedge \cdots \\ & \wedge \left( \theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ =& \int _ X \left( \theta _1+\theta _1'+\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}+\mathrm{dd}^{\mathrm{c}}P_{\theta _1'}[\varphi _1']_{\mathcal{I}} \right)\wedge \left( \theta _2+\mathrm{dd}^{\mathrm{c}}P_{\theta _2}[\varphi _2]_{\mathcal{I}}\right)\wedge \cdots \\ & \wedge \left( \theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ =& \operatorname{vol}(T_1,T_2,\ldots ,T_ n)+\operatorname{vol}(T_1',T_2,\ldots ,T_ n). \end{aligned} \]
4 This follows from (18.1.22), proposition 16.6.27 1 and proposition 13.11.4 6.
5 Our assertion follows from the monotonicity theorem theorem 13.13.11.
$\square$
In view of
18.1.23 2, we may extend the definition of mixed volumes to the more general setting as follows:
Definition 18.1.25.7731FA7D8BFD Let $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ for $i=1,\ldots ,n$. Write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for each $i=1,\ldots ,n$. Take a Kähler form $\omega $ on $X$. We define the mixed volume $\operatorname{vol}(T_1,\ldots ,T_ n)$ as
0141EC0ADB1E
18.1.26
\begin{equation} \operatorname{vol}\left(T_1,\ldots ,T_ n\right)=\lim _{\epsilon \to 0+}\operatorname{vol}\left(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega \right). \end{equation}
Note that $\operatorname{vol}(T_1+\epsilon \omega ,\ldots ,T_ n+\epsilon \omega )$ is decreasing as $\epsilon \to 0+$ due to 18.1.23 3, so the limit exists.
Furthermore, $\operatorname{vol}(T_1,\ldots ,T_ n)$ does not depend on the choice of $\omega $, as a consequence of 18.1.23 3.
Proposition 18.1.27.DF8A923E8F3E
74EAB3B1411B
The mixed volume is symmetric in its $n$-variables.
AEBD6B8A6D3F
Consider closed positive $(1,1)$-currents $T_1,\ldots ,T_ n,T_1'$ on $X$, then
7.22
\begin{equation} \operatorname{vol}(T_1+T_1',T_2,\ldots ,T_ n)=\operatorname{vol}(T_1,T_2,\ldots ,T_ n)+\operatorname{vol}(T_1',T_2,\ldots ,T_ n). \end{equation}
439CA74101C3
Consider closed positive $(1,1)$-currents $T_1,\ldots ,T_ n$ on $X$ and $\lambda \geq 0$, then
\[ \operatorname{vol}(\lambda T_1,T_2,\ldots ,T_ n)=\lambda \operatorname{vol}(T_1,T_2,\ldots ,T_ n). \]
CB1FF7FB0BEC
Suppose that $T_1,\ldots ,T_ n,S_1,\ldots ,S_ n$ are closed positive $(1,1)$-currents on $X$ such that $T_ i\preceq _{\mathcal{I}} S_ i$ and $\{ T_ i\} =\{ S_ i\} $ for each $i=1,\ldots ,n$. Then
7.23
\begin{equation} \operatorname{vol}(T_1,\ldots ,T_ n)\leq \operatorname{vol}(S_1,\ldots ,S_ n). \end{equation}
5BDABA896DC6
Suppose that $T_1,\ldots ,T_ n$ are closed positive $(1,1)$-currents on $X$. Then
7.24
\begin{equation} \operatorname{vol}\left(T_1,\ldots ,T_ n \right)=\operatorname{vol}\left(\operatorname{Reg}T_1,\ldots ,\operatorname{Reg}T_ n \right). \end{equation}
EF34C0318EDD
For any closed positive $(1,1)$-current $T$ on $X$, we have
1BAF68507D97
18.1.28
\begin{equation} \operatorname{vol}(T,\ldots ,T)=\operatorname{vol}T. \end{equation}
Proof.
1, 2, 3, 4 are immediate consequences of the corresponding properties proved in 18.1.23.
5 Using 3 and 4, it suffices to establish the following: Suppose that
5EAAF326E66A
18.1.29
\begin{equation} T=\sum _ i c_ i[E_ i] \end{equation}
is a closed positive $(1,1)$-current on $X$, where $\{ E_ i\} $ is a countable collection of prime divisors on $X$ and $c_ i>0$. Then
A35A4555844E
18.1.30
\begin{equation} \operatorname{vol}\left(T,T_2,\ldots ,T_ n \right)=0 \end{equation}
for any closed positive $(1,1)$-currents $T_2,\ldots ,T_ n$ on $X$.
Step 1. We first assume that (18.1.29) is a finite sum. Fix a Kähler form $\omega $ on $X$.
In this case, thanks to 3 and 4 again, we may assume that $T=[E]$ for some prime divisor $E$ on $X$. Write $[E]=\theta _{\varphi }$. Then $\varphi $ has analytic singularities thanks to proposition 5.20.4. Therefore, $P_{\theta +\epsilon \omega }[\varphi ]_{\mathcal{I}}\sim \varphi $ for any $\epsilon >0$ due to proposition 16.6.22. Write $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for $i=2,\ldots ,n$, and choose $C>0$ so that $C\omega +\theta _ i$ are Kähler forms for each $i=2,\ldots ,n$. Then
\[ \begin{aligned} & \operatorname{vol}(T,T_2,\ldots ,T_ n)\\ =& \lim _{\epsilon \to 0+} \int _ X \left([E]+\epsilon \omega \right)\wedge \left( \theta _2+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _2+\epsilon \omega }[\varphi _2]_{\mathcal{I}}\right)\wedge \cdots \\ & \wedge \left( \theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _ n+\epsilon \omega }[\varphi _ n]_{\mathcal{I}}\right)\\ =& \lim _{\epsilon \to 0+} \epsilon \int _ X \omega \wedge \left( \theta _2+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _2+\epsilon \omega }[\varphi _2]_{\mathcal{I}}\right)\wedge \cdots \\ & \wedge \left( \theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta _ n+\epsilon \omega }[\varphi _ n]_{\mathcal{I}}\right)\\ \leq & \lim _{\epsilon \to 0+} \epsilon \int _ X \omega \wedge \left( \theta _2+C\omega \right)\wedge \cdots \wedge \left( \theta _ n+C\omega \right)\\ =& 0, \end{aligned} \]
where the inequality step follows from the monotonicity theorem theorem 13.13.11 and the fact that $P_{\theta _ i+\epsilon \omega }[\varphi _ i]_{\mathcal{I}}\leq 0$ for all $i=2,\ldots ,n$.
Step 2. We prove the case where $\{ E_ i\} $ is infinite. We may assume that $i$ runs over $\mathbb {Z}_{>0}$.
Write $T_ i=\theta _{i}+\mathrm{dd}^{\mathrm{c}}\varphi _ i$ for $i=2,\ldots ,n$ as before. Fix a Kähler form $\omega $ on $X$ so that $\theta _ i+\omega $ is a Kähler form for each $i=2,\ldots ,n$. Fix $\epsilon >0$.
We can find $N_0>0$ so that for any $N\geq N_0$, the class of
\[ \epsilon \omega + \sum _{i=N+1}^{\infty }c_ i[E_ i] \]
is Kähler. Take a Kähler form $\omega _ N$ in this class. Then the currents
\[ \sum _{i=1}^ N c_ i [E_ i]+\omega _ N,\quad \sum _{i=1}^{\infty } c_ i [E_ i]+\epsilon \omega \]
all lie in the same cohomology class. Moreover,
\[ \sum _{i=1}^{\infty } c_ i [E_ i]+\epsilon \omega \preceq _{\mathcal{I}}\sum _{i=1}^ N c_ i [E_ i]+\omega _ N. \]
Therefore, by the monotonicity in 5, followed by the finite divisorial case of Step 1,
\[ \begin{aligned} \operatorname{vol}\left(T,T_2,\ldots ,T_ n\right)\leq & \operatorname{vol}\left(T+\epsilon \omega ,T_2,\ldots ,T_ n\right)\\ \leq & \operatorname{vol}\left(\sum _{i=1}^ N c_ i [E_ i]+\omega _ N,T_2,\ldots ,T_ n\right)\\ =& \operatorname{vol}\left(\omega _ N,T_2,\ldots ,T_ n\right)\\ \leq & \int _ X \omega _ N\wedge (\theta _2+\omega )\wedge \cdots \wedge (\theta _ n+\omega ). \end{aligned} \]
Letting $N\to \infty $, and using $\{ \omega _ N\} \to \{ \epsilon \omega \} $, we get
\[ \operatorname{vol}\left(T,T_2,\ldots ,T_ n\right)\leq \epsilon \int _ X \omega \wedge (\theta _2+\omega )\wedge \cdots \wedge (\theta _ n+\omega ). \]
Since $\epsilon >0$ is arbitrary, we conclude (18.1.30).
6 Write $T=\theta _{\varphi }$. In more concrete terms, we need to show that
\[ \lim _{\epsilon \to 0+}\int _ X (\theta +\epsilon \omega +\mathrm{dd}^{\mathrm{c}}P_{\theta +\epsilon \omega }[\varphi ]_{\mathcal{I}})^ n=\int _ X (\theta +\mathrm{dd}^{\mathrm{c}}P_{\theta }[\varphi ]_{\mathcal{I}})^ n. \]
We may replace $\varphi $ by $P_{\theta }[\varphi ]_{\mathcal{I}}$ and assume that $\varphi $ is $\mathcal{I}$-model in $\mathrm{PSH}(X,\theta )$. Then we claim that
\[ \varphi =\inf _{\epsilon >0}P_{\theta +\epsilon \omega }[\varphi ]_{\mathcal{I}}. \]
From this, our assertion follows from proposition 16.4.4.
The $\leq $ direction is clear. For the converse, since $\varphi $ is $\mathcal{I}$-model, it suffices to show that for each prime divisor $E$ over $X$, we have
\[ \nu (\varphi ,E)\leq \nu \left(\inf _{\epsilon >0}P_{\theta +\epsilon \omega }[\varphi ]_{\mathcal{I}},E\right). \]
We simply compute
\[ \nu \left(\inf _{\epsilon >0}P_{\theta +\epsilon \omega }[\varphi ]_{\mathcal{I}},E\right)\geq \sup _{\epsilon >0}\nu \left(P_{\theta +\epsilon \omega }[\varphi ]_{\mathcal{I}},E\right)=\nu (\varphi ,E). \]
$\square$
Lemma 18.1.31.1AEDF3A60767 Let $\omega $ be a Kähler form on $X$. Then there is a constant $C>0$ depending only on $X,\omega ,\{ T_1\} ,\ldots ,\{ T_ n\} $ such that
\[ 0\leq \operatorname{vol}(T_1+\epsilon \omega ,\ldots , T_ n+\epsilon \omega )-\operatorname{vol}(T_1,\ldots ,T_ n)\leq C\epsilon \]
for any $\epsilon \in [0,1]$.
Proof.
By the multi-linearity established in proposition 18.1.27, we can write
\[ \operatorname{vol}\left(T_1+\epsilon \omega ,\ldots , T_ n+\epsilon \omega \right)-\operatorname{vol}(T_1,\ldots ,T_ n) \]
as a linear combination of the $2^ n-1$-terms:
\[ \epsilon ^ m\operatorname{vol}\left(\omega ^ m, T_{i_1},\ldots ,T_{i_{n-m}} \right) \]
where $J\subseteq \{ 1,\ldots ,n\} $ is a non-empty subset of cardinality $m$, and $\{ i_1,\ldots ,i_{n-m}\} $ is the complement of $J$. It suffices to show that the coefficients are bounded by a constant as above. Take $D\geq 0$ as the minimal non-negative number so that $D\{ \omega \} +\{ T_ i\} $ is nef for each $i=1,\ldots ,n$. Then by the monotonicity theorem theorem 13.13.11,
\[ \operatorname{vol}\left(\omega ^ m, T_{i_1},\ldots ,T_{i_{n-m}} \right)\leq \{ \omega \} ^ m\cap \left( \{ T_{i_1}\} +D\{ \omega \} \right) \cap \cdots \cap \left( \{ T_{i_{n-m}}\} +D\{ \omega \} \right). \]
Our assertion follows.
$\square$
Next we show that theorem
16.9.57 continues to hold even when $\varphi $ has vanishing mass. We prove a slightly more general result:
Theorem 18.1.32.A320A575BBEB Let $\theta _1,\ldots ,\theta _ n$ be smooth closed real $(1,1)$-forms representing big cohomology classes. Suppose that $(\varphi _ j^ k)_{k\in I}$ are nets in $\mathrm{PSH}(X,\theta _ j)$ and $\varphi _ j\in \mathrm{PSH}(X,\theta _ j)$ for $j=1,\ldots ,n$. We assume that $\varphi _ j^ k\xrightarrow {d_ S}\varphi _ j$ for each $j=1,\ldots ,n$. Then
BCAEFDCBCBE3
18.1.33
\begin{equation} \lim _{k\in I}\operatorname{vol}\left( \theta _{1,\varphi _1^ k},\ldots , \theta _{n,\varphi _ n^ k}\right)=\operatorname{vol}\left( \theta _{1,\varphi _1},\ldots , \theta _{n,\varphi _ n}\right). \end{equation}
Proof.
Fix a Kähler form $\omega $. Then for any $\epsilon >0$, we have
\[ \lim _{k\in I}\operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1^ k},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n^ k}\right)=\operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n}\right) \]
as a consequence of corollary 16.9.36, theorem 16.9.30 and theorem 16.9.49.
Now thanks to 18.1.31, there is $C>0$ so that for each $k\in I$,
\[ \begin{aligned} 0\leq & \operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1^ k},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n^ k}\right)-\operatorname{vol}\left( \theta _{1,\varphi _1^ k},\ldots , \theta _{n,\varphi _ n^ k}\right)\leq C\epsilon ,\\ 0\leq & \operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n}\right)-\operatorname{vol}\left( \theta _{1,\varphi _1},\ldots , \theta _{n,\varphi _ n}\right)\leq C\epsilon . \end{aligned} \]
Therefore, (18.1.33) follows.
$\square$
Corollary 18.1.34.3305393E4400 Let $\theta $ be a smooth closed real $(1,1)$-form representing a big cohomology class. Let $(\varphi _ j)_{j\in I}$ be a net in $\mathrm{PSH}(X,\theta )$ and $\varphi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi _ j\xrightarrow {d_ S}\varphi $. Then
55BF10D53948
18.1.35
\begin{equation} \operatorname{vol}\theta _{\varphi _ j}\to \operatorname{vol}\theta _{\varphi },\quad \int _ X \theta _{\varphi _ j}^ n\to \int _ X \theta _{\varphi }^ n. \end{equation}
Proof.
The first part of (18.1.35) is a special case of theorem 18.1.32, while the second part of (18.1.35) is a special case of theorem 16.9.30.
$\square$
The mixed volume has a log-concavity property:
Proposition 18.1.36.BB7441317AAF Let $T_1,\ldots ,T_ n$ be closed positive $(1,1)$-currents on $X$. Then
\[ \operatorname{vol}(T_1,\ldots ,T_ n)\geq \prod _{i=1}^ n(\operatorname{vol}T_ i)^{1/n}. \]
Proof.
We may assume that $\operatorname{vol}T_ i>0$ for each $i=1,\ldots ,n$ since there is nothing to prove otherwise. In this case, we need to show that
\[ \begin{split} & \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}P_{\theta _1}[\varphi _1]_{\mathcal{I}}\right)\wedge \cdots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}P_{\theta _ n}[\varphi _ n]_{\mathcal{I}}\right)\\ \geq & \prod _{i=1}^ n\left(\int _ X \left(\theta _ i+\mathrm{dd}^{\mathrm{c}}P_{\theta _ i}[\varphi _ i]_{\mathcal{I}} \right)^ n \right)^{1/n}. \end{split} \]
This is a special case of theorem 13.13.5.
$\square$
Next, we also study how the mixed volume behaves under bimeromorphic transforms.
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.
$\square$
As for pushforward, we also have a similar result. We need a preliminary result:
Lemma 18.1.38.30F0E1037DAE Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a Kähler manifold $Y$. Then for any non-divisorial closed positive $(1,1)$-current $T$ on $Y$, we have
\[ \pi ^* \pi _* T=T+\sum _{i=1}^ N c_ i [E_ i] \]
for finitely many $\pi $-exceptional divisors $E_ i$ and $c_ i>0$.
Proof.
Let $E$ be the exceptional locus of $\pi $. Then
\[ T=\mathds {1}_{Y\setminus E}\pi ^* \pi _* T. \]
Therefore,
\[ \pi ^* \pi _* T-T=\mathds {1}_ E\pi ^* \pi _* T, \]
which has the stated form, due to the support theorems, see [Section 8, DemBook].
$\square$
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.
$\square$
A particular corollary of corollary
18.1.39 will be useful later.
Corollary 18.1.41.1F0FA3B61EBC Let $\pi \colon X\rightarrow Z$ be a proper bimeromorphic morphism from $X$ to a Kähler manifold $Z$. Assume that $T$ is an $\mathcal{I}$-good closed positive $(1,1)$-current on $X$. Then so is $\pi _*T$.
Proof.
We may assume that $\int _ X T^ n>0$. Then by corollary 18.1.39,
\[ \operatorname{vol}\pi _*T=\operatorname{vol}T>0 \]
as well. Since $T$ is $\mathcal{I}$-good, we have
\[ \operatorname{vol}T=\int _ X T^ n. \]
But $\int _ X T^ n=\int _ Z (\pi _*T)^ n$, so
\[ \operatorname{vol}\pi _*T=\int _ Z (\pi _*T)^ n>0. \]
It follows that $\pi _*T$ is $\mathcal{I}$-good.
$\square$
Finally we establish a semicontinuity property of the mixed volumes.
Theorem 18.1.42.763BE634A054 Let $(\varphi _ i^ j)_{j\in J}$ ($i=1,\ldots ,n$) be nets in $\mathrm{PSH}(X,\theta _ i)$, and $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ ($i=1,\ldots ,n$). Assume that for each prime divisor $E$ over $X$, we have
\[ \lim _{j\in J} \nu \left(\varphi _ i^ j,E\right)=\nu \left(\varphi _ i,E\right),\quad i=1,\ldots ,n. \]
Then
\[ \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right) \leq \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n\right). \]
Proof.
Step 1. We first assume that $\operatorname{vol}(\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i^ j)>0$ and $\operatorname{vol}(\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i)>0$ for all $i=1,\ldots ,n$ and $j\in J$.
Without loss of generality, we may assume that the $\varphi _ i^ j$'s and the $\varphi _ i$'s are $\mathcal{I}$-model for all $i=1,\ldots ,n$ and $j\in J$. Our assertion becomes
055370851A43
18.1.43
\begin{equation} \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right) \leq \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1\right) \wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n\right). \end{equation}
For each $j\in J$, define
\[ \psi _ i^ j\coloneqq \operatorname *{sup}_{k\geq j}{}\! {}^{*} \varphi _ i^ k,\quad i=1,\ldots ,n. \]
Observe that $\psi _ i^ j$ is $\mathcal{I}$-good thanks to proposition 18.1.19, since after the preceding reduction the tail family is bounded above by $0$. It follows from corollary 5.16.9 and our assumption that
\[ \lim _{j\in J}\nu \left(\psi _ i^ j,E\right)=\nu \left(\varphi _ i,E\right),\quad i=1,\ldots ,n. \]
Moreover, for each $j\in J$, the same formula gives $\varphi _ i\preceq _{\mathcal{I}}\psi _ i^ j$. Since $\varphi _ i$ is $\mathcal{I}$-model and $\psi _ i^ j$ is $\mathcal{I}$-good, proposition 16.8.17 gives
\[ \varphi _ i\leq P_{\theta _ i}\left[\psi _ i^ j\right]. \]
For each $i=1,\ldots ,n$, we define
\[ \psi _ i=\inf _{j\in J}P_{\theta _ i}\left[\psi _ i^ j\right]. \]
In particular, $\psi _ i\geq \varphi _ i$, so $\int _ X\theta _{i,\psi _ i}^ n>0$. Since $\psi _ i^ j$ is $\mathcal{I}$-good, $P_{\theta _ i}[\psi _ i^ j]$ is both model and $\mathcal{I}$-model. Due to proposition 16.6.29, $\psi _ i$ is $\mathcal{I}$-model. Thanks to proposition 16.4.4, we know
\[ \nu (\psi _ i,E)=\nu (\varphi _ i,E) \]
for any $i=1,\ldots ,n$ and any prime divisor $E$ over $X$. In other words, $\psi _ i\sim _{\mathcal{I}} \varphi _ i$ for $i=1,\ldots ,n$. But both $\varphi _ i$ and $\psi _ i$ are $\mathcal{I}$-good, therefore,
\[ \psi _ i\sim _{P} \varphi _ i,\quad i=1,\ldots ,n. \]
By proposition 16.8.22, we have
\[ \int _ X (\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1)\wedge \cdots \wedge (\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n)=\int _ X (\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1)\wedge \cdots \wedge (\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n). \]
Next by proposition 16.8.22 again,
\[ \begin{split} & \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right)\\ \leq & \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n^ j \right). \end{split} \]
On the other hand, due to proposition 16.4.4 and corollary 16.9.14, for each $i=1,\ldots ,n$, we have
\[ \psi _ i^ j\xrightarrow {d_ S}\psi _ i. \]
We conclude from theorem 16.9.30 that
\[ \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n^ j \right)=\int _ X (\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1)\wedge \cdots \wedge (\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n). \]
Putting these equations together, (18.1.43) follows.
Step 2. Next we handle the general case.
Fix a Kähler form $\omega $ on $X$. For any $\epsilon \in (0,1]$, from Step 1, we know that
\[ \begin{split} & \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right) \\ \leq & \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n\right). \end{split} \]
Using 18.1.31, we have
\[ \begin{aligned} & \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right)\\ \leq & \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right)\\ \leq & \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n\right)\\ \leq & \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n\right)+C\epsilon . \end{aligned} \]
But since $\epsilon $ is arbitrary, our assertion follows.
$\square$
Pome
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