Lemma 11.31.10.BD07AFC2CACB Let $(X,\omega )$ be a compact Hermitian manifold of dimension $n\geq 2$, and let $G$ be a positive $(n-1,n-1)$-current with $\mathrm{dd}^{\mathrm{c}}G=0$. Put $\mu =\omega \wedge G$ and $M=\mu (X)$. If $a,b$ are smooth real functions and $C\geq 0$ is a constant with $\mathrm{dd}^{\mathrm{c}}a,\mathrm{dd}^{\mathrm{c}}b\geq -C\omega $, then
\[ 0\leq \int _ X\mathrm{d}(a-b)\wedge \mathrm{d}^ c(a-b)\wedge G \leq 2C M\| a-b\| _\infty . \]
For every smooth real function $v$, the total variation norms of the order-zero currents $\mathrm{d}v\wedge G$ and $\mathrm{d}^ cv\wedge G$ satisfy
\[ \| \mathrm{d}v\wedge G\| _{\mathrm{TV}}+ \| \mathrm{d}^ cv\wedge G\| _{\mathrm{TV}} \leq C_\omega M^{1/2} \left(\int _ X\mathrm{d}v\wedge \mathrm{d}^ cv\wedge G\right)^{1/2}. \]
Here the total variation norms are computed with respect to $\omega $, and $C_\omega $ depends only on $(X,\omega )$.
Proof.
Set $v=a-b$. The identity $\mathrm{dd}^{\mathrm{c}}(v^2)=2\mathrm{d}v\wedge \mathrm{d}^ cv+2v\mathrm{dd}^{\mathrm{c}}v$, integrated against $G$, and the equation $\mathrm{dd}^{\mathrm{c}}G=0$ give
\[ \int _ X\mathrm{d}v\wedge \mathrm{d}^ cv\wedge G =-\int _ Xv\mathrm{dd}^{\mathrm{c}}v\wedge G. \]
The measures $(\mathrm{dd}^{\mathrm{c}}a+C\omega )\wedge G$ and $(\mathrm{dd}^{\mathrm{c}}b+C\omega )\wedge G$ are positive, and each has mass $CM$, because the terms $\mathrm{dd}^{\mathrm{c}}a\wedge G$ and $\mathrm{dd}^{\mathrm{c}}b\wedge G$ integrate to zero. Their difference $\mathrm{dd}^{\mathrm{c}}v\wedge G$ therefore has total variation at most $2CM$, which proves the upper bound. The lower bound holds because $\mathrm{d}v\wedge \mathrm{d}^ cv=(\mathrm{i}/2\pi )\partial v\wedge \bar\partial v$ is a positive $(1,1)$-form.
For the second estimate, work in a local unitary frame for $\omega $. Positivity of $G$ means that its coefficients form a positive semidefinite matrix of measures. These entries are absolutely continuous with respect to the trace measure $\mu $: the diagonal entries are dominated by the trace, and the off-diagonal entries by the Cauchy–Schwarz inequality for positive semidefinite matrices. The resulting matrix of densities has trace bounded by a constant depending only on the normalization. Pairing $\mathrm{d}v\wedge G$ or $\mathrm{d}^ cv\wedge G$ with a test one-form $\eta $, and applying the matrix Cauchy–Schwarz inequality and then the integral Cauchy–Schwarz inequality, we obtain
\[ |\langle \mathrm{d}v\wedge G,\eta \rangle | +|\langle \mathrm{d}^ cv\wedge G,\eta \rangle | \leq C_\omega \| \eta \| _\infty M^{1/2} \left(\int _ X\mathrm{d}v\wedge \mathrm{d}^ cv\wedge G\right)^{1/2}. \]
A finite cover by such frames and a partition of unity give the global estimate. Taking the supremum over smooth test forms of norm at most one yields the total variation bound. When $M=0$, positivity gives $G=0$, and both estimates hold trivially.
$\square$
Pome
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