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The proofs of the standard results recalled in this section can be found in [BEGZ10].
Let $X$ be a complex manifold.
Definition 13.11.1.1A52175C3D64 Let $\varphi _1,\ldots ,\varphi _ p\in \mathrm{PSH}(X)$. We set
\[ O_ k \coloneqq \bigcap _{j=1}^ p \{ \varphi _ j>-k\} ,\quad k\in \mathbb {N}. \]
We say that $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined if for each connected open subset $U\subseteq X$, any smooth positive $(1,1)$-form $\omega $ on $U$, and each compact subset $K\subseteq U$, we have
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13.11.2
\begin{equation} \sup _{k\geq 0} \int _{K\cap O_ k}\left.\left(\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}(\varphi _ j\lor (-k))\right)\right|_ U\wedge \omega ^{\dim U-p}<\infty . \end{equation}
In this case, we define the non-pluripolar product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ by
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13.11.3
\begin{equation} \mathds {1}_{O_ k}\, \mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p=\mathds {1}_{O_ k}\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\left( \varphi _ j\lor (-k)\right) \end{equation}
on $\bigcup _{k\geq 0} O_ k$ and extend it by zero to $X$.
As recalled in section
5.15, an $\mathcal{F}$-open subset means an open subset with respect to the plurifine topology.
Proposition 13.11.4.9E81D36B1290 Let $\varphi _1,\ldots ,\varphi _ p\in \mathrm{PSH}(X)$.
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The product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is local with respect to the plurifine topology in the following sense: Let $O\subseteq X$ be an $\mathcal{F}$-open subset and $\psi _1,\ldots ,\psi _ p\in \mathrm{PSH}(X)$. Assume that
\[ \varphi _ j|_{O}=\psi _ j|_{O},\quad j=1,\ldots ,p, \]
and that
\[ \bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j\textup{ and } \bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\psi _ j \]
are both well-defined, then
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13.11.5
\begin{equation} \left.\left(\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j\right)\right|_{O}=\left.\left(\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\psi _ j\right)\right|_{O}. \end{equation}
If furthermore $O$ is open in the usual topology, then the product
\[ \bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j|_{O} \]
on $O$ is well-defined and
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13.11.6
\begin{equation} \left. \left(\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j\right)\right|_{O}=\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j|_{O}. \end{equation}
Let $\mathcal{U}$ be an open covering of $X$. Then $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined if and only if each product
\[ \bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j|_{U},\quad U\in \mathcal{U}. \]
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The current $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ and the fact that it is well-defined depend only on the currents $\mathrm{dd}^{\mathrm{c}}\varphi _ j$, not on the choice of the $\varphi _ j$'s nor on the ordering of the $\varphi _ j$'s.
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When $\varphi _1,\ldots ,\varphi _ p\in L^{\infty }_{\mathrm{loc}}(X)$, the non-pluripolar product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined and is equal to the Bedford–Taylor product. More generally, if $q\leq p$, and we only assume that $\varphi _1,\ldots ,\varphi _ q\in L^{\infty }_{\mathrm{loc}}(X)$, and that the non-pluripolar product $T\coloneqq \mathrm{dd}^{\mathrm{c}}\varphi _{q+1}\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined, then the non-pluripolar product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is also well-defined, and is equal to the Bedford–Taylor product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ q\wedge T$ as in 13.10.2.
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Assume that $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined. Then $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ puts no mass on pluripolar sets.
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Assume that $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is well-defined. Then $\bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j$ is a closed positive $(p,p)$-current on $X$.
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The product is multilinear: Let $\psi _1\in \mathrm{PSH}(X)$, $a,b>0$ then
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13.11.7
\begin{equation} \mathrm{dd}^{\mathrm{c}}(a\varphi _1+b\psi _1)\wedge \bigwedge _{j=2}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j =a\, \mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \bigwedge _{j=2}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j + b\, \mathrm{dd}^{\mathrm{c}}\psi _1\wedge \bigwedge _{j=2}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j \end{equation}
in the sense that the left-hand side is well-defined if and only if both terms on the right-hand side are well-defined, and the equality holds in that case.
Proof. For bounded potentials, Bedford--Taylor plurifine locality identifies their products on every plurifine open set on which the potentials agree. Apply it to the bounded truncations $\max\{\varphi_j,-k\}$ and $\max\{\psi_j,-k\}$, on the intersection of $O$ with the finite-level loci of all the potentials. The truncated products agree there. Let $k\to\infty$ in the increasing restriction measures defining the non-pluripolar product. The omitted pole sets have zero mass by construction, so equality holds on all of $O$. This limiting argument, including invariance under changes of local potentials, is established in the local calculus of non-pluripolar products.
In view of
3, we do not need to specify whether our product $\mathrm{dd}^{\mathrm{c}}\varphi _1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p$ is the Bedford–Taylor product or the non-pluripolar product when the $\varphi _ i$'s are all locally bounded.
Example 13.11.8.00950EDCBA2C Let $\varphi \in \mathrm{PSH}(X)$, then
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13.11.9
\begin{equation} (\mathrm{dd}^{\mathrm{c}}\varphi )^1=\mathds {1}_{\{ \varphi >-\infty \} }\, \mathrm{dd}^{\mathrm{c}}\varphi . \end{equation}
Here $(\mathrm{dd}^{\mathrm{c}}\varphi )^1$ on the left-hand side denotes the non-pluripolar product, while $\mathrm{dd}^{\mathrm{c}}\varphi $ on the right-hand side denotes the usual derivative in the sense of current.
The problem is local, so we may assume that $X=\Delta ^ n$, the unit polydisk in $\mathbb {C}^ n$. Take $f\in C^{\infty }_ c(\Delta ^ n)$ with $f\geq 0$. We need to show that
\[ \lim _{k\to \infty } \int _{\Delta ^ n} \mathds {1}_{\{ \varphi >-k\} }f\, \mathrm{dd}^{\mathrm{c}}\bigl (\varphi \lor (-k)\bigr )\wedge \omega ^{n-1}=\int _{\Delta ^ n} \mathds {1}_{\{ \varphi >-\infty \} }f\, \mathrm{dd}^{\mathrm{c}}\varphi \wedge \omega ^{n-1}, \]
where $\omega $ is the standard Kähler form on $\mathbb {C}^ n$. By the dominated convergence theorem, this reduces immediately to the $1$-dimensional case:
\[ \lim _{k\to \infty } \int _{\Delta ^1} \mathds {1}_{\{ \varphi >-k\} }f\, \Delta \bigl (\varphi \lor (-k)\bigr )=\int _{\Delta ^1} \mathds {1}_{\{ \varphi >-\infty \} }f\, \Delta \varphi , \]
for any subharmonic function $\varphi $ on the unit disk $\Delta ^1$, which is a special case of [Proposition 3.3, EK24].
Definition 13.11.10.05C17B17762F Let $T_1,\ldots ,T_ p$ be closed positive $(1,1)$-currents on $X$. We say that $T_1\wedge \cdots \wedge T_ p$ is well-defined if there exists an open covering $\mathcal{U}$ of $X$ such that on each $U\in \mathcal{U}$, we can find $\varphi _ j^ U\in \mathrm{PSH}(U)$ ($j=1,\ldots ,p$) such that
\[ \mathrm{dd}^{\mathrm{c}}\varphi _ j^ U=T_ j,\quad j=1,\ldots ,p \]
and $\mathrm{dd}^{\mathrm{c}}\varphi _1^ U\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p^ U$ is well-defined. In this case, we define the non-pluripolar product $T_1\wedge \cdots \wedge T_ p$ as the closed positive $(p,p)$-current on $X$ defined by
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13.11.11
\begin{equation} \left(T_1\wedge \cdots \wedge T_ p\right)|_ U=\mathrm{dd}^{\mathrm{c}}\varphi _1^ U\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}\varphi _ p^ U,\quad U\in \mathcal{U}. \end{equation}
The product $T_1\wedge \cdots \wedge T_ p$ is independent of the choices we made thanks to proposition
13.11.4 1 and
2.
proposition 13.11.4 can be formulated in terms of currents without any difficulty.
Proposition 13.11.14.FB457F67F566 Let $X$ be a compact Kähler manifold and let $T_1,\ldots ,T_ p$ be closed positive $(1,1)$-currents on $X$. Then $T_1\wedge \cdots \wedge T_ p$ is well-defined.
Proof. Write $T_j=\theta_j+\mathrm{dd}^{\mathrm c}\varphi_j$ by the global potential lemma, and fix a Kähler form $\omega$. Choose $C$ so large that every $\tau_j=\theta_j+C\omega$ is semipositive. The compact mass estimate proves
\[\int_X\langle T_1\wedge\cdots\wedge T_p\rangle\wedge\omega^{n-p}\leq\int_X\tau_1\wedge\cdots\wedge\tau_p\wedge\omega^{n-p}<\infty.\]
Thus the increasing restricted truncation measures have locally finite coefficient measures. The construction therefore defines a current everywhere; closedness of the truncation limit and the local-calculus theorem show that it is independent of the chosen potentials and patches between charts. Compactness and the Kähler form are used in the displayed uniform mass bound.
This proposition explains why we usually work in the setting of compact Kähler manifolds.
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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13.11.18
\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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