Proposition 5.28.23EDAB69A7289

Proof. On the intersection of two coordinate balls, the two corrected potentials differ by a smooth function. The rule $\mathrm{dd}^{\mathrm{c}}(p+f)=\mathrm{dd}^{\mathrm{c}}p+\mathrm{dd}^{\mathrm{c}}f$, for bounded $p$ and smooth $f$ in bounded local products, follows by smooth approximation; together with the binomial formula, it shows that the two expansions agree. The same argument handles a smooth change in the decomposition of a current. Moreover, any two decompositions of the same $(1,1)$-current differ by a smooth function: the difference of the potentials has a smooth Hessian, and taking its local Euclidean trace and applying elliptic regularity shows that it is smooth. Symmetry, multilinearity and locality then follow term by term. Bounded Bedford–Taylor products wedged with smooth forms have locally finite variation and do not charge pluripolar sets, so these properties pass to the finite expansion as well.

It remains to prove positivity, which does not follow from the signs of the summands. Write the factors as $T_ i=\theta _ i+\mathrm{dd}^{\mathrm{c}}u_ i$, $1\leq i\leq k$, where each $\theta _ i$ is a smooth real $(1,1)$-form and each $u_ i$ is locally bounded and quasi-plurisubharmonic, and assume $T_ i\geq 0$. Fix $\varepsilon >0$, a point of the coordinate ball and a positive constant form $\kappa $. On a sufficiently small neighborhood of the point, each $\theta _ i$ is close to a constant, hence closed, form $\theta _ i'$:

Then $S_ i=\theta _ i'+\varepsilon \kappa +\mathrm{dd}^{\mathrm{c}}u_ i\geq T_ i$ is a positive closed current there, with a bounded local plurisubharmonic potential, and

where the last term is a smooth semipositive form. Expanding the product of the currents $T_ i+2\varepsilon \kappa $ therefore gives a sum of positive currents, so this product is positive near the chosen point. Such neighborhoods, although they depend on $\varepsilon $, cover the coordinate ball, so the product of the $T_ i+2\varepsilon \kappa $ is positive on the whole ball. By multilinearity, it converges weakly to the product of the $T_ i$ as $\varepsilon \to 0$, which proves positivity. $\square$