Pome

Lemma 5.29.10EF54C7C9A0A8

Lemma 5.29.10.EF54C7C9A0A8 Let $(Y,\Omega )$ be compact Hermitian, let $B'\Subset B\subset Y$ be coordinate balls, and let $p\in \mathrm{PSH}(B)\cap C^{0,\alpha }(B)$ have norm at most $N$, with $0<\alpha \leq 1$. There are $c,C>0$, uniform for these references, such that

\[ (\mathrm{dd}^{\mathrm{c}}p)^ n\bigl (B'\cap \{ v<-t\} \bigr )\leq C\mathrm e^{-ct} \quad (t\geq 0) \]

for every $v\in \mathrm{PSH}(Y,\Omega )$ with $\sup _ Yv=0$.

Proof. Choose a smooth quadratic $Q$ with $\mathrm{dd}^{\mathrm{c}}Q\geq \Omega $ on $B$ and a fixed constant $D$ such that $w=v+Q-D\leq 0$. For integers $m\geq 1$, set

\[ a_ m=\max (w,-m),\qquad b_ m=\max (w,-m-1),\qquad f_ m=a_ m-b_ m. \]

These are bounded plurisubharmonic truncations; $0\leq f_ m\leq 1$, it vanishes on $\{ w\geq -m\} $, and it equals one on $\{ w<-m-1\} $. By lemma 5.26.2 on $(Y,\Omega )$ and the bounded correction $Q-D$,

\[ \| f_ m\| _{L^1(B)}\leq C\mathrm e^{-am} \]

for a fixed $a>0$. Use the proof of lemma 5.29.7, with $M=m+1$, keeping its estimate (5.29.8) before optimization. Taking $\varepsilon =\mathrm e^{-am/(\alpha +2n)}$ gives

\[ \int \chi f_ m(\mathrm{dd}^{\mathrm{c}}p)^ n \leq C(1+m)^2\exp \left(-\frac{a\alpha m}{\alpha +2n}\right) \leq C\mathrm e^{-cm} \]

for some $c>0$. Large $m$ ensure that the convolution radius is admissible; the common local mass handles the other integers. The bounded difference between $w$ and $v$, and then choosing an integer just below the given height, prove the estimate for every real $t\geq 0$. $\square$

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