PomeLemma 8.7.6.E7EA08EFFDE7 Every continuous admissible $u$ satisfies $\int _ X H_ m(u)>0$.
Lemma 8.7.6E7EA08EFFDE7
Proof. Suppose this integral is zero. For $s>0$ put $w=\max (u,\min _ Xu+s)$ and $U=\{ u<\min _ Xu+s\} $. The open set $U$ is nonempty, $0\leq w-u\leq s$, and $w=u$ outside it. Set $\sigma _ u=\omega +\mathrm{dd}^{\mathrm{c}}u$, $\sigma _ w=\omega +\mathrm{dd}^{\mathrm{c}}w$, and $\mu _ q=\sigma _ u^ q\wedge \sigma _ w^{m-q}\wedge \omega ^{n-m}$. The contact lemma gives $\mathbf1_{X\setminus U}\mu _ q\geq \mathbf1_{X\setminus U}\mu _ m$, for $0\leq q\leq m$. For $q<m$ the telescoping identity is
\[ \mu _ q-\mu _ m=\mathrm{dd}^{\mathrm{c}}(w-u)\wedge T_ q,\qquad T_ q=\sigma _ u^ q\wedge \sum _{r=0}^{m-q-1}\sigma _ w^ r\wedge \sigma _ u^{m-q-1-r} \wedge \omega ^{n-m}. \]
Integration by parts on $X$ therefore gives
\[ \int _ U(\mu _ q-\mu _ m)\leq \int _ X(\mu _ q-\mu _ m) =\int _ X(w-u)\mathrm{dd}^{\mathrm{c}}T_ q. \]
This identity follows first for smooth approximants and then by uniform product continuity. The derivatives of $T_ q$ contain only fixed metric derivatives and products of at most $m-1$ admissible factors, so the absolute-form lemma makes $\mathrm{dd}^{\mathrm{c}}T_ q$ a measure of uniformly controlled variation. Uniform convergence also justifies its continuous weight. On $U$, locality gives $\sigma _ w=\omega $ and thus
\[ |\mathrm{dd}^{\mathrm{c}}T_ q|\leq C\sum _{r=0}^{m-1}\mu _ r. \]
The constant depends only on the metric and $m,n$, and not on $u$ or $s$. Consequently
\[ \sum _{q=0}^ m\int _ U\mu _ q \leq (m+1)\int _ U H_ m(u)+Cms\sum _{r=0}^{m-1}\int _ U\mu _ r. \]
For sufficiently small $s$, absorption and $H_ m(u)=0$ force all these nonnegative integrals to vanish. But $\mu _0=\omega ^ n$ on the nonempty open set $U$, a contradiction. $\square$
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