Proposition 8.12.5BB9187C844DA

Proof. Suppose instead that $H_ m(u)=0$. Fix $s>0$, set $c=\inf _ Xu+s$, and put $w=\max \{ u,c\} $. The upper semicontinuity of $u$ makes $U=\{ u<c\} $ open, and the definition of the infimum makes it nonempty. Moreover $0\leq w-u\leq s$ and this difference is zero outside $U$. The maximum is bounded admissible with its canonical pointwise values by 8.2.9.

For $0\leq q\leq m$, define the positive Radon measures

Thus $\mu _ m=H_ m(u)=0$. For $q<m$, let $T_ q$ be the sum of $m-q$ currents with $q+r$ slots equal to $u$ and $m-q-1-r$ slots equal to $w$, for $0\leq r<m-q$, each of degree $m-1$ and wedged with $\omega ^{n-m}$. In smooth notation this is

Here each displayed product denotes the symmetric shifted current $R_{m-1}$ of (8.9.5), wedged with $\omega ^{n-m}$.

Choose smooth bounded admissible $u_ j\downarrow u$ and the regularized maxima $w_ j=M_{\epsilon _ j}(u_ j,c)\downarrow w$, where $\epsilon _ j\downarrow 0$, as in 8.2.9. Both lists have a common bound. The smooth polynomial identity and Stokes' theorem on the compact manifold give

Joint decreasing convergence passes the left side to its limit. The preceding lemma, with $F(x,y)=y-x$ and the decreasing lists $u_ j,w_ j$, passes the right side to its limit. Hence

The difference of the two decreasing sequences need not itself decrease; the continuous two-variable weight is essential here.

On the open set $U$, the potential $w$ is constant. Locality of the raw recursion therefore sets every factor $\mathrm{dd}^{\mathrm{c}}w$ to zero there, so that its shifted factors become $\omega $. The formula in the preceding lemma expresses $\mathrm{dd}^{\mathrm{c}}T_ q|_ U$ as finitely many smooth metric-derivative forms wedged with powers of $\omega +\mathrm{dd}^{\mathrm{c}}u$ of degrees at most $m-1$. Use smooth admissible approximations to $u$ and the constant $c$ in these local expressions; their limits are the same restricted currents. For those smooth approximations, the absolute mixed-form estimate 8.2.7 bounds each such term by a constant times the corresponding positive trace, and therefore by $C\sum _{r=0}^{m-1}\mu _ r$ on $U$ after taking joint limits. More explicitly, a term with $r$ factors is bounded by $C(\omega +\mathrm{dd}^{\mathrm{c}}u)^ r\omega ^{n-r}$; this is $C\mu _ r$ on $U$. Against compactly supported nonnegative tests in $U$, weak convergence and lower semicontinuity of total variation preserve the estimate. Exhaustion gives the measure inequality

Here $C$ depends only on $\omega ,m,n$, not on $u$ or $s$; a fixed finite coordinate cover bounds all the smooth metric coefficients.

Since $w-u$ vanishes outside $U$, positivity gives

Summing these $m$ inequalities and choosing $s>0$ with $mCs<1$ forces $\sum _{r=0}^{m-1}\int _ U\mu _ r=0$. But locality gives $\mu _0=\omega ^ n$ on the nonempty open set $U$, whose volume is positive. This contradiction proves the result. $\square$