Proposition 20.3.59AA3908FEC2B4

Proof. The curvature current of $a$ is the sum of the pullbacks of the two positive currents $A\eta +\mathrm{dd}^{\mathrm{c}}w$ and $\omega _ Z+\mathrm{dd}^{\mathrm{c}}h$, which proves positivity.

Suppose first that both potentials are bounded. Expanding the bounded product by multilinearity, only the terms with one factor from $C$ and $k$ factors from $Z$ survive: two factors from $C$ vanish for dimension reasons, and so do $k+1$ factors from $Z$. The binomial coefficient is $k+1$, which gives the formula for bounded potentials once the exterior product $q^*(A\eta +\mathrm{dd}^{\mathrm{c}}w)\wedge r^*(\omega _ Z+\mathrm{dd}^{\mathrm{c}}h)^ k$ is identified with the product measure $(A\eta +\mathrm{dd}^{\mathrm{c}}w)\otimes (\omega _ Z+\mathrm{dd}^{\mathrm{c}}h)^ k$. To justify this identification at the present regularity, add local plurisubharmonic corrections on each factor and regularize by decreasing convolution on smaller product charts. For the smooth approximants the identity is the exterior product of forms. Local Bedford–Taylor convergence, together with the finite expansion in the smooth background, gives weak convergence of the factor measures, with uniformly bounded variation on compact subcharts. The product measures then converge weakly: one first tests against products of continuous functions on the two factors and then uses uniform approximation by finite sums of such products. This proves the identity in the limit.

For general potentials, subtract constants so that $w,h\leq 0$, and set $w_ j=\max (w,-j)$ and $h_ j=\max (h,-j)$. On

the sum agrees with the bounded potential $w_ j\circ q+h_ j\circ r$. This set is plurifine open and Borel, and it lies in $\{ a>-2j\} $. Locality of the products of bounded truncations therefore identifies the restriction to $R_ j$ of the extended measure of $a$ with that of the bounded sum. By the bounded case, this is $k+1$ times the product of the restrictions of the truncated measures, which are exactly the restrictions of the two extended measures of the factors.

The sets $R_ j$ increase to the product of the complements of the two pole sets, and both sides of (20.3.60) give zero mass to the remaining set: for the left-hand side it is the pole set of $a$, and for the right-hand side this follows because each factor measure charges no pluripolar set and the product measures are $\sigma $-finite. Monotone convergence of positive measures proves the formula, even when the total mass is infinite. $\square$