Proposition 16.14.6CB0B256B92D7

Proof. The potentials $\rho _ j$ exist by lemma 6.14.2. By lemma 16.14.4, all $u_ j$ belong to $W^{2,2}_{\mathrm{loc}}(U)\subset W^{2,1}_{\mathrm{loc}}(U)$, so the matrices $A_ j$ and $B_ j$ are defined almost everywhere, and the positive current $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}u_ j=\mathrm{dd}^{\mathrm{c}}(\rho _ j+u_ j)$ has coefficients in $L^2_{\mathrm{loc}}$.

We first show that $A_ j$ is positive semidefinite almost everywhere. For each $\xi \in \mathbb C^ n$ the distribution $\sum _{a,b}(\rho _ j+u_ j)_{a\bar b}\xi _ a\bar\xi _ b$ is positive, because $T_ j$ is a positive current; since it is a locally integrable function, it is nonnegative almost everywhere, by proposition 2.5.3(3),(4) and lemma 2.5.1(2) (its mollifications are nonnegative and converge to it in $L^1_{\mathrm{loc}}$). We apply this to the countably many $\xi $ with rational real and imaginary parts. Outside the union of the corresponding null sets, $\sum _{a,b}(A_ j)_{ab}\xi _ a\bar\xi _ b\geq 0$ for all such $\xi $, hence for all $\xi \in \mathbb C^ n$, because these $\xi $ are dense and a quadratic form on $\mathbb C^ n$ is continuous.

Next we bound the coefficients. By the trace identity (10.2.13),

which is bounded on $W$, since $Lu_ j$ is locally bounded and $\overline W$ is compact. Since $T_ j=\frac{\mathrm i}{2\pi }\sum _{a,b}(A_ j)_{ab}\, \mathrm dz_ a\wedge \mathrm d\bar z_ b$, the function $\Lambda T_ j$ is the trace $\sum _{a,b}h^{a\bar b}(A_ j)_{ab}=\operatorname {tr}(H^{-1}A_ j)$ of $A_ j$ relative to $h$, as in (10.2.14), where $H=(h_{a\bar b})$. As $\overline W$ is compact, $H\leq CI$ on $W$ for a constant $C$, so $H^{-1}\geq C^{-1}I$; since $A_ j$ is positive semidefinite, $\operatorname {tr}(H^{-1}A_ j)=\operatorname {tr}(A_ j^{1/2}H^{-1}A_ j^{1/2})\geq C^{-1}\operatorname {tr}A_ j$ almost everywhere on $W$. For a positive semidefinite Hermitian matrix every diagonal entry is at most the trace, and every entry satisfies $|a_{ab}|^2\leq a_{aa}a_{bb}$, by the positivity of the principal $2\times 2$ minors. Therefore $A_ j$ is bounded on $W$.

The functions $\rho _ j+u_ j$ are continuous and plurisubharmonic on $W$, and they have bounded complex Hessian in the sense of section 7.13. The first identity is therefore the first part of proposition 7.13.4. The functions $\rho _ j+f_ j$ are real Borel functions in $W^{2,1}_{\mathrm{loc}}(W)$, and adding $\rho _ j$ to both $u_ j$ and $f_ j$ does not change the contact set $\{ u_ j=f_ j\} $. Hence the second part of proposition 7.13.4 gives the second identity. Moreover proposition 7.13.3, applied to $\rho _ j+u_ j$ and $\rho _ j+f_ j$, shows that $A_ j=B_ j$ almost everywhere on $\{ u_ j=f_ j\} $, hence on $E$. Consequently $\mathbf1_ ED(B_1,\ldots ,B_ n)=\mathbf1_ ED(A_1,\ldots ,A_ n)$ almost everywhere. This density is nonnegative, since the mixed discriminant of positive semidefinite matrices is nonnegative (lemma 5.3.3(3)), and it is bounded, since $D$ is a polynomial in the entries of the bounded matrices $A_ j$. $\square$