Corollary 13.11.15.AE7DB6D35BE9 Let $X$ and $Z$ be connected compact Kähler manifolds of dimensions $n$ and $m$, respectively, and let $p_1\colon X\times Z\to X$ and $p_2\colon X\times Z\to Z$ be the projections. Let $\theta $ be a closed real smooth $(1,1)$-form on $X$, let $\phi \in \mathrm{PSH}(X,\theta )$, and let $\gamma $ be a closed smooth positive $(1,1)$-form on $Z$. Put
\[ Y=X\times Z,\qquad \Theta =p_1^*\theta +p_2^*\gamma ,\qquad \Phi =p_1^*\phi . \]
Suppose that $W\in \mathrm{PSH}(Y,\Theta )$ and $\Phi -C\leq W\leq \Phi +C$ for some $C>0$. Set
\[ T=\Theta +\mathrm{dd}^{\mathrm{c}}W,\qquad S=\Theta +\mathrm{dd}^{\mathrm{c}}\Phi , \]
and write $W_ z(x)=W(x,z)$. Then $W_ z\in \mathrm{PSH}(X,\theta )$ for every $z\in Z$. For $0\leq j\leq n$ and every smooth real $(m,m)$-form $\eta $ on $Z$,
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13.11.16
\begin{equation} \begin{aligned} & \int _ Y(W-\Phi )p_2^*\eta \wedge T^ j\wedge S^{n-j}\\ & \qquad =\int _ Z\left(\int _ X(W_ z-\phi )\, \theta _{W_ z}^ j\wedge \theta _\phi ^{n-j}\right)\eta (z), \end{aligned} \end{equation}
and
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13.11.17
\begin{equation} \int _ Y p_2^*\eta \wedge T^ n =\int _ Z\left(\int _ X\theta _{W_ z}^ n\right)\eta (z). \end{equation}
Proof.
By linearity, it suffices to consider $\eta \geq 0$. All the non-pluripolar products that occur below are well-defined by proposition 13.11.14. We work on a relatively compact product chart $Q\times D\subseteq X\times Z$ and let $a$ and $b$ be smooth local potentials of $p_1^*\theta $ and $p_2^*\gamma $. After subtracting constants, put $w=W+a+b\leq 0$ and $f=\Phi +a+b\leq 0$. For $k\geq 1$, set
\[ \begin{aligned} w_ k& =w\lor (-k),& f_ k& =f\lor (-k),\\ O_ k& =\{ w>-k\} \cap \{ f>-k\} ,& \mu _{j,k}& =\mathds {1}_{O_ k}(\mathrm{dd}^{\mathrm{c}}w_ k)^ j\wedge (\mathrm{dd}^{\mathrm{c}}f_ k)^{n-j}. \end{aligned} \]
For $z\in D$, write $w_ z(x)=w(x,z)$ and $f_ z(x)=f(x,z)$, and set
\[ \mu _{j,k,z} =\mathds {1}_{\{ w_ z>-k\} \cap \{ f_ z>-k\} } \bigl (\mathrm{dd}^{\mathrm{c}}_ x(w_ z\lor (-k))\bigr )^ j\wedge \bigl (\mathrm{dd}^{\mathrm{c}}_ x(f_ z\lor (-k))\bigr )^{n-j}. \]
Apply 13.10.5 with $u_1=\cdots =u_ j=w_ k$, $u_{j+1}=\cdots =u_ n=f_ k$, and
\[ h=\rho G\mathds {1}_{O_ k},\qquad \rho \in \mathcal{C}^{\infty }_ c(Q\times D),\quad \rho \geq 0, \]
where $G$ is any bounded non-negative Borel function. This gives
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\begin{equation} \int _{Q\times D}\rho G\, p_2^*\eta \wedge \mu _{j,k} =\int _ D\left(\int _ Q\rho _ zG_ z\, \mu _{j,k,z}\right)\eta (z). \end{equation}
By plurifine locality, $\mu _{j,k}$ increases to the local expression of $T^ j\wedge S^{n-j}$, while, for every $z\in D$, $\mu _{j,k,z}$ increases to the local expression of $\theta _{W_ z}^ j\wedge \theta _\phi ^{n-j}$. Taking $G=\delta +C\geq 0$ in (13.11.18), applying monotone convergence, subtracting the corresponding constant-weight identity, and summing over a non-negative partition of unity proves (13.11.16). The same argument with constant weight and degree $n$ proves (13.11.17).
$\square$
Pome
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