Proposition 13.11.49E81D36B1290

Proposition 13.11.4.9E81D36B1290 Let $\varphi _1,\ldots ,\varphi _ p\in \mathrm{PSH}(X)$.

1. 4C9D393253D0

   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

   C3F3FF3F504B 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

   7E91D53462AC 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

   \[ \bigwedge _{j=1}^ p \mathrm{dd}^{\mathrm{c}}\varphi _ j|_{U},\quad U\in \mathcal{U}. \]
2. 13DFC0FA2988

   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.

3. 1353BEBCB3BB

   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.

4. F4627B44F775

   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.

5. 90ED277A28AD

   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 $X$.¹

6. 87188EAC5CDB

   The product is multilinear: Let $\psi _1\in \mathrm{PSH}(X)$, $a,b>0$ then

   A1F77BA03A26 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.