Corollary 13.14.23.BED3510B9193 Suppose $v_+(\omega )=+\infty $. Let $(\varepsilon _ j)_{j\geq 1}$ and $(b_ j)_{j\geq 1}$ be real sequences with $0<\varepsilon _ j<1$, $\varepsilon _ j\to 0$, and $b_ j>\int _ X\omega ^ n$. There are smooth potentials $w_ j\in \mathrm{PSH}(X,\omega )$ with $\sup _ Xw_ j=0$ such that
\[ \begin{gathered} \omega +\mathrm{dd}^{\mathrm{c}}w_ j\geq (1-\varepsilon _ j)\omega , \qquad \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}w_ j)^ n=b_ j, \\ \| \mathrm{dd}^{\mathrm{c}}w_ j\| _{L^1(\gamma )} \leq 2M\varepsilon _ j. \end{gathered} \]
Moreover, $w_ j\to 0$ in $L^1(\mathrm{d}V)$ and $\mathrm{e}^{-pw_ j}\to 1$ in $L^1(\mathrm{d}V)$ for every fixed $p>0$. The full Gauduchon trace measures converge to $\omega \wedge \gamma ^{n-1}$ in total variation.
Proof.
By proposition 13.14.15, we can choose a smooth $u_ j\in \mathrm{PSH}(X,\omega )$ with $\sup _ Xu_ j=0$ and
\[ \int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u_ j)^ n>\varepsilon _ j^{-n}b_ j. \]
Put $T_ j=\omega +\mathrm{dd}^{\mathrm{c}}u_ j$. The polynomial
\[ F_ j(s)=\int _ X\bigl ((1-s)\omega +sT_ j\bigr )^ n, \qquad 0\leq s\leq \varepsilon _ j, \]
satisfies $F_ j(0)<b_ j$, and also $F_ j(\varepsilon _ j)\geq \varepsilon _ j^ n\int _ XT_ j^ n>b_ j$ by positivity of the mixed terms. The intermediate value theorem gives $s_ j\in (0,\varepsilon _ j)$ with $F_ j(s_ j)=b_ j$. Set $w_ j=s_ ju_ j$. Then $\sup _ Xw_ j=0$, $\omega +\mathrm{dd}^{\mathrm{c}}w_ j=(1-s_ j)\omega +s_ jT_ j\geq (1-\varepsilon _ j)\omega $, and $w_ j$ has the prescribed mass $b_ j$.
The Gauduchon identity gives $\| T_ j\| _{L^1(\gamma )}=\| \omega \| _{L^1(\gamma )}=M$. Hence
\[ \| \mathrm{dd}^{\mathrm{c}}w_ j\| _{L^1(\gamma )} =s_ j\| T_ j-\omega \| _{L^1(\gamma )} \leq 2M\varepsilon _ j. \]
Since the absolute value of the trace of a Hermitian matrix is bounded by the trace of its absolute value, the same bound holds for the total variation of $\mathrm{dd}^{\mathrm{c}}w_ j\wedge \gamma ^{n-1}$. With $K_\omega $ as in (5.26.1), we also have $\| w_ j\| _{L^1(\mathrm{d}V)}\leq K_\omega \varepsilon _ j$.
Finally, let $\alpha ,C$ be the constants of lemma 5.26.2 for $(X,\omega ,\mathrm{d}V)$, and fix $p>0$. For all sufficiently large $j$, one has $ps_ j\leq \alpha $. Convexity of the exponential function, applied with weight $ps_ j/\alpha $, gives
\[ 0\leq \mathrm{e}^{-pw_ j}-1 \leq \frac{ps_ j}{\alpha }\bigl (\mathrm{e}^{-\alpha u_ j}-1\bigr ). \]
Integrating, we bound the $L^1(\mathrm{d}V)$ norm of $\mathrm{e}^{-pw_ j}-1$ by $pC\varepsilon _ j/\alpha $, which proves the last convergence.
$\square$
Pome
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