Proposition 7.13.41466731348C1

Proof. For smooth $u_ j$ we polarize (7.2.4), as in the proof of corollary 7.8.8. Applied to the smooth function $\sum _ jt_ ju_ j$ with $t_1,\ldots ,t_ n\in \mathbb R$, it gives the identity

between exterior powers of smooth forms, and both sides are polynomials in $t_1,\ldots ,t_ n$. We compare the coefficients of $t_1\cdots t_ n$. Since $2$-forms commute, on the left this coefficient is $n!$ times the exterior product $\mathrm{dd}^{\mathrm{c}}u_1\wedge \cdots \wedge \mathrm{dd}^{\mathrm{c}}u_ n$, which is the Bedford–Taylor product by proposition 7.2.2(1); on the right it is $(n!)^2\pi ^{-n}D(A_1,\dots ,A_ n)\, \mathrm dV_{2n}$, by the definition of the mixed discriminant. This proves the first formula for smooth $u_ j$.

For continuous $u_ j$, let $u_{j,\varepsilon }=u_ j*\rho _\varepsilon $ be their radial mollifications, which are smooth and plurisubharmonic on $\Omega _\varepsilon $ (proposition 5.9.2). They converge to $u_ j$ uniformly on compact sets (lemma 2.5.1), so their products converge weakly by proposition 7.2.6. By proposition 2.5.3, the complex Hessian matrix of $u_{j,\varepsilon }$ is the mollification $A_{j,\varepsilon }$ of the matrix $A_ j$ of locally bounded functions. Let $K\subset \Omega $ be compact. For small $\varepsilon $, the entries of $A_{j,\varepsilon }$ are bounded on $K$ by the bounds of the entries of $A_ j$ on a fixed compact neighborhood of $K$, and by lemma 2.5.1 they converge to those of $A_ j$ in $L^1(K)$. Products of functions which are bounded on $K$ uniformly in $\varepsilon $ and converge in $L^1(K)$ converge in $L^1(K)$, because $|ab-a'b'|\leq |a||b-b'|+|b'||a-a'|$. Since $D$ is a polynomial in the matrix entries, it follows that $D(A_{1,\varepsilon },\dots ,A_{n,\varepsilon })\to D(A_1,\dots ,A_ n)$ in $L^1(K)$. By the first part, $(n!/\pi ^ n)D(A_{1,\varepsilon },\dots ,A_{n,\varepsilon })\, \mathrm dV_{2n}$ is the product of the $\mathrm{dd}^{\mathrm{c}}u_{j,\varepsilon }$, so the weak limit of these products is $(n!/\pi ^ n)D(A_1,\dots ,A_ n)\, \mathrm dV_{2n}$. This proves the first formula in general.

By lemma 7.13.2, each $u_ j$ belongs to $W^{2,2}_{\mathrm{loc}}(\Omega )\subset W^{2,1}_{\mathrm{loc}}(\Omega )$, so proposition 7.13.3, applied to $u_ j$ and the chosen representative of $v_ j$, gives $A_ j=B_ j$ almost everywhere on $\{ u_ j=v_ j\} $, hence on $E$. Substituting into the first formula proves the second, and also shows that $\mathbf1_ ED(B_1,\dots ,B_ n)=\mathbf1_ ED(A_1,\dots ,A_ n)$ is locally bounded. $\square$