Proposition 14.24.525FABE38E60B

Proof. Locally, expand $T$ into powers of the smooth form $\omega $ multiplied by closed Bedford–Taylor products of $\mathrm{dd}^{\mathrm{c}}u$ and $\mathrm{dd}^{\mathrm{c}}v$. The operator $\mathrm{dd}^{\mathrm{c}}$ then acts only on the powers of $\omega $, and the local mass bounds in section 5.27 show that $\mathrm{dd}^{\mathrm{c}}T$ is a finite signed measure.

We first construct $E$ and prove (14.24.6). On a coordinate ball, convolve $u$, $v$ and the coefficients of $\omega $ with a common nonnegative radial mollifier. Write the resulting smooth positive forms as $A_\rho =\omega _\rho +\mathrm{dd}^{\mathrm{c}}u_\rho $ and $B_\rho =\omega _\rho +\mathrm{dd}^{\mathrm{c}}v_\rho $, and let $T_\rho $ be their mixed sum. On a slightly larger ball, add the same fixed multiple of $|z|^2$ to each potential to obtain plurisubharmonic functions. Their radial regularizations decrease to the originals, so Bedford–Taylor continuity applies to all closed Hessian products. Since $u,v$ are continuous, $u_\rho ,v_\rho ,w_\rho $ converge uniformly on compact subsets.

For a closed smooth product $S_\rho $ appearing in $T_\rho $, the identity

has a distributional limit: both Hessian products on the right converge weakly with uniformly bounded local variation, and their weights converge uniformly. The same holds after multiplication by the convergent smooth factors coming from $\omega _\rho $. Thus the positive measures $E_\rho =\mathrm{d} w_\rho \wedge \mathrm d^{\mathrm c}w_\rho \wedge T_\rho $ converge to a finite positive measure $E$. By (14.24.9), this limit does not depend on the regularization, so the local limits agree on overlapping charts.

For smooth regularizations, the chain rule gives

The last current tends to zero, because $\mathrm{dd}^{\mathrm{c}}w_\rho \wedge T_\rho =A_\rho ^ n-B_\rho ^ n$ tends weakly to $A^ n-B^ n=0$ with uniformly bounded variation, and $\chi '(w_\rho )$ converges uniformly. Passing to the limit and integrating by parts on the compact manifold $X$ gives (14.24.6). A partition of unity justifies this passage without a global regularization; the terms containing derivatives of the cutoff functions cancel after summation.

Let $\sigma =w_*(\mathrm{dd}^{\mathrm{c}}T)$ and $\nu =w_*E$. Equation (14.24.6) says that $\nu ''=\sigma $ as distributions on $\mathbb R$. Stokes' theorem and $\mathrm{dd}^{\mathrm{c}}w\wedge T=0$ give

Consequently $k(t)=\int (s-t)_+\, \mathrm{d}\sigma (s)$ is globally Lipschitz and vanishes outside $[a,b]$. It satisfies $k''=\sigma $, so the compactly supported distribution $\nu -k(t)\, \mathrm{d} t$ has second derivative zero and therefore vanishes. Positivity of $\nu $ and continuity of $k$ imply $k\geq 0$. This proves (14.24.7), and the assertion about level sets follows because $\nu $ is absolutely continuous.

We now prove (14.24.8). On a coordinate neighborhood of $\overline V$, set $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$ and choose $c_ V>0$ with $F\omega ^ n\geq c_ V\beta ^ n$. Let $A_{\mathrm{ac}}$ denote the absolutely continuous part of the current $A$, whose coefficients are locally integrable functions. Applying lemma 5.28.10 with the constant function $g=c_ V^{1/n}$ and differentiating the coefficient measures, we obtain $\det _\beta A_{\mathrm{ac}}\geq c_ V$ almost everywhere. The local Sobolev regularity in section 5.27 gives $w_\rho \to w$ strongly in $W^{1,2}_{\mathrm{loc}}$, so a subsequence of the gradients converges almost everywhere. Moreover, $A_\rho \to A_{\mathrm{ac}}$ almost everywhere, by differentiation of the coefficient measures.

Since $T_\rho $ contains the term $A_\rho ^{n-1}$ and all its terms are positive, Fatou's lemma, applied to $E_\rho $, yields the measure inequality

The cofactor matrix of $A_{\mathrm{ac}}$ has smallest eigenvalue at least $\det A_{\mathrm{ac}}/\operatorname {tr}A_{\mathrm{ac}}\geq c_ V/\operatorname {tr}A_{\mathrm{ac}}$, where traces and determinants are taken with respect to $\beta $. The trace is locally integrable because $A$ has locally finite mass. The chain rule for Sobolev functions, (14.24.10) and the Cauchy–Schwarz inequality give

The last factor is finite and is absorbed into $C_ V$. The values of $p'$ at the finitely many corners of $p$ are immaterial, because $E$ charges no level set of $w$. $\square$