Lemma 5.28.7.6C81352D9EE7 Let $T_ i=\vartheta _ i+\mathrm{dd}^{\mathrm{c}}u_ i\geq 0$, $i=1,2$, where the $\vartheta _ i$ are smooth real $(1,1)$-forms on $U$ and the $u_ i$ are locally bounded quasi-plurisubharmonic functions on $U$. If
\[ T_ i^ n\geq F_ i\beta ^ n, \qquad F_ i\geq 0\text{ measurable}, \]
then, for every integer $0\leq k\leq n$,
C06268DF902D
5.28.8
\begin{equation} T_1^ k\wedge T_2^{n-k} \geq F_1^{k/n}F_2^{(n-k)/n}\beta ^ n. \end{equation}
Consequently, for $a,b\geq 0$,
4547B4E0976D
5.28.9
\begin{equation} (aT_1+bT_2)^ n \geq \bigl (aF_1^{1/n}+bF_2^{1/n}\bigr )^ n\beta ^ n. \end{equation}
Proof.
Fix $W\Subset U$. Since the forms $\vartheta _ i$ are smooth, there is a constant $C$ such that, for every $x\in W$ and every small ball $B_ r(x)$, the constant forms
\[ A_ i=\vartheta _ i(x)+Cr\beta \]
satisfy $0\leq S_ i\leq 2Cr\beta $ on $B_ r(x)$, where $S_ i=A_ i-\vartheta _ i$. If $q_ i$ is a quadratic potential for $A_ i$, then $u_ i+q_ i$ is plurisubharmonic on $B_ r(x)$ and
\[ \bigl (\mathrm{dd}^{\mathrm{c}}(u_ i+q_ i)\bigr )^ n=(T_ i+S_ i)^ n\geq F_ i\beta ^ n. \]
Apply theorem 5.27.5 to these two closed Hessians. After expansion, every term other than $T_1^ k\wedge T_2^{n-k}$ contains some $S_ i$ and is therefore bounded by $C'r\mu $, where
\[ \mu =\sum _{\ell +m\leq n-1}T_1^\ell \wedge T_2^ m\wedge \beta ^{n-\ell -m} \]
is a locally finite measure.
A finite cover and a partition of unity give, on $W$,
\[ T_1^ k\wedge T_2^{n-k} \geq F_1^{k/n}F_2^{(n-k)/n}\beta ^ n-C'r\mu . \]
Letting $r\downarrow 0$ proves (5.28.8), and the binomial expansion then gives (5.28.9).
$\square$
Pome
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