PomeLemma 5.29.12.1017EA20E8D8 For every compact $K\subset \mathbb P^ n$,
\[ T(K)\leq \mathrm e\exp \bigl (-\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)^{-1/n}\bigr ). \]
The right-hand side is zero for capacity zero.
5.29.11 Capacity bounds for Hölder references510FD09C306A
To pass from the exponential tail estimate to a capacity estimate, we use a closed background on an auxiliary projective space. The local capacity comparison then returns the result to the original Hermitian manifold.
For the Fubini–Study form $\omega _{\mathrm{FS}}$ on $\mathbb P^ n$, normalized by $\int \omega _{\mathrm{FS}}^ n=1$, define for a compact set $K$
\[ T(K)=\inf _{v\in \mathrm{PSH}(\mathbb P^ n,\omega _{\mathrm{FS}}),\ \sup v=0} \exp (\sup _ Kv). \]
We recall the classical global extremal-function theorem in this closed setting [GZ17]. If $K$ is non-pluripolar, the upper semicontinuous regularization of
\[ V_ K=\sup \{ v\in \mathrm{PSH}(\mathbb P^ n,\omega _{\mathrm{FS}}):v|_ K\leq 0\} \]
is bounded, nonnegative and $\omega _{\mathrm{FS}}$-plurisubharmonic. Its Monge–Ampère probability is supported on $K$. If $K$ is pluripolar, $\sup V_ K=+\infty $ and $T(K)=0$.
Proof. The pluripolar case follows from the recalled theorem and the characterization of capacity-zero sets. Otherwise let $M=\sup V_ K^*<\infty $. Subtracting $\sup _ Kv$ from a normalized competitor, and conversely normalizing competitors for $V_ K$, shows directly that $T(K)=\mathrm e^{-M}$.
If $M\geq 1$, the function $q=V_ K^*/M$ satisfies $0\leq q\leq 1$ and is an admissible capacity test. Since the background is closed,
\[ 1=\int _ K\omega _{V_ K^*}^ n \leq M^ n\int _ K\omega _ q^ n \leq M^ n\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K). \]
Here $\omega _{V_ K^*}\leq M\omega _ q$ follows from $M\geq 1$. Thus $M\geq \operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)^{-1/n}$. If $M<1$, $V_ K^*$ itself is a capacity test, so the same supported probability gives $\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)=1$. Since $T(K)\leq 1$, the stated factor $\mathrm e$ covers this case. $\square$
Proposition 5.29.13.D00B685B1791 Let $(X,\omega )$ be compact Hermitian. Fix $0<\alpha \leq 1$ and $N>0$. There are $a,C>0$ such that every $u\in \mathrm{PSH}(X,\omega )\cap C^{0,\alpha }(X)$ with $\| u\| _{C^{0,\alpha }}\leq N$ satisfies
\[ \omega _ u^ n(E)\leq C\exp \bigl (-a\operatorname {Cap}_\omega (E)^{-1/n}\bigr ) \quad (E\subset X\text{ Borel}). \]
Consequently, for each $s>0$ there is a uniform $C_ s$ with $\omega _ u^ n(E)\leq C_ s\operatorname {Cap}_\omega (E)^ s$.
Proof. Fix nested coordinate balls $B'\Subset B$ and a smooth quadratic $Q$ with $\mathrm{dd}^{\mathrm{c}}Q\geq \omega $ on $B$. The function $p=u+Q$ is plurisubharmonic with a common Hölder bound, and $\omega _ u^ n\leq (\mathrm{dd}^{\mathrm{c}}p)^ n$ there. Regard $B$ as a ball in the affine chart $\mathbb C^ n\subset \mathbb P^ n$ and restrict this local measure to $B'$, extending it by zero. By lemma 5.29.10 on $(\mathbb P^ n,\omega _{\mathrm{FS}})$, it has a uniform exponential tail on every normalized global Fubini–Study potential. Its total mass is uniformly bounded by local Chern–Levine–Nirenberg estimates.
For a non-pluripolar compact $K\subset B'$, choose a normalized potential $v$ with $\sup _ Kv\leq \log T(K)+1$. For small $T(K)$ the tail estimate gives
\[ (\mathrm{dd}^{\mathrm{c}}p)^ n(K)\leq C T(K)^ c \leq C\exp \bigl (-c\operatorname {Cap}_{\omega _{\mathrm{FS}}}(K)^{-1/n}\bigr ). \]
The common mass bound handles the remaining $T(K)$, and pluripolar sets have zero local product mass. A bounded local potential for $\omega _{\mathrm{FS}}$ turns every global test $0\leq v\leq 1$ into a plurisubharmonic function of a fixed bounded range on $B$. Hence
\[ \operatorname {Cap}_{\omega _{\mathrm{FS}}}(K) \leq C_1\operatorname {cap}(K,B) \leq C_2\operatorname {Cap}_\omega (K), \]
where the last inequality is the gluing construction of lemma 5.29.2 on the original manifold. These inequalities have the required direction after taking negative inverse powers. Inner regularity of the finite local product extends the bound from compact to Borel subsets of $B'$. Summing over a fixed finite cover proves the first assertion. The second follows by bounding $x^{-s}\exp (-ax^{-1/n})$ on $0<x\leq \operatorname {Cap}_\omega (X)$. $\square$
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