Lemma 10.7.6.D51F6B990A39 Let $M$ be a complex manifold of dimension $n\geq 1$ with a Hermitian form $\omega $ and the volume form $\mathrm dV=\omega ^ n/n!$, and let $F\to M$ be a holomorphic line bundle with a smooth quadratic metric $h$ such that $c_1(F,h)\geq a\omega $ for a constant $a>0$. Suppose that $M$ carries a Kähler form $\sigma \geq \omega /2$ and functions $\chi _ R$ with the properties stated in lemma 10.7.5. Let $g$ be an $F$-valued $(n,1)$-form with $\bar\partial g=0$ in the sense of distributions and $\int _ M|g|_{\omega ,h}^2\, \mathrm dV<\infty $. Then there is an $F$-valued $(n,0)$-form $u$ with $\bar\partial u=g$ and
71ED94BE6393
10.7.7
\begin{equation} \int _ M|u|_{\omega ,h}^2\, \mathrm dV \leq \frac1{2\pi a}\int _ M|g|_{\omega ,h}^2\, \mathrm dV. \end{equation}
Proof.
Let $B_\sigma =[2\pi c_1(F,h),\Lambda _\sigma ]$. Since $c_1(F,h)$ is positive definite, $B_\sigma $ is positive definite on $(n,1)$-forms by lemma 10.7.1(1). Let $T=D''\colon L^2_{(n,0)}\to L^2_{(n,1)}$ and $S=D''\colon L^2_{(n,1)}\to L^2_{(n,2)}$ be the operators with maximal distributional domains, for the norms defined by $\sigma $ and $h$. By lemma 10.7.4, applied with $h_0=h_1=h_2=h$, they are closed and densely defined, $ST=0$, and compactly supported smooth forms are dense in $\operatorname{Dom}T^*\cap \operatorname{Dom}S$ for the graph norm $\| v\| +\| T^*v\| +\| Sv\| $.
Let $v$ be a compactly supported smooth $(n,1)$-form. We integrate the pointwise Cauchy–Schwarz inequality $|\langle g,v\rangle _\sigma |^2\leq \langle B_\sigma ^{-1}g,g\rangle _\sigma \langle B_\sigma v,v\rangle _\sigma $ for the positive operator $B_\sigma $ and use (10.7.3), in which $D''{}^*v=T^*v$ (lemma 10.7.4) and $D''v=Sv$. This gives
\[ \Bigl |\int _ M\langle g,v\rangle _\sigma \, \mathrm dV_\sigma \Bigr |^2 \leq \int _ M\langle B_\sigma ^{-1}g,g\rangle _\sigma \, \mathrm dV_\sigma \cdot \bigl (\| T^*v\| _\sigma ^2+\| Sv\| _\sigma ^2\bigr ). \]
By lemma 10.7.1(3), the density $\langle B_\sigma ^{-1}g,g\rangle _\sigma \, \mathrm dV_\sigma $ does not depend on $\sigma $. Computing it with $\omega $ in place of $\sigma $, and using that $2\pi c_1(F,h)\geq 2\pi a\omega $, so that $\langle [2\pi c_1(F,h),\Lambda _\omega ]^{-1}g,g\rangle _{\omega ,h}\leq (2\pi a)^{-1}|g|_{\omega ,h}^2$ by the same lemma, we find that the first integral is at most the right side of (10.7.7).
The form $g$ lies in $\ker S$, and in $L^2_{(n,1)}$ for $\sigma $ and $h$, because $\sigma \geq \omega /2$ gives $|g|_{\sigma ,h}^2\, \mathrm dV_\sigma \leq 2|g|_{\omega ,h}^2\, \mathrm dV$, again by lemma 10.7.1(3). Hence both sides of the displayed inequality converge along a graph-convergent sequence, and by the density of compactly supported smooth forms the inequality extends to all $v\in \operatorname{Dom}T^*\cap \operatorname{Dom}S$. Theorem 6.2.4 therefore gives $u\in \operatorname{Dom}T$ with $Tu=g$ and $\| u\| _\sigma ^2$ at most the first integral. Finally, $|u|_{\sigma ,h}^2\, \mathrm dV_\sigma =|u|_{\omega ,h}^2\, \mathrm dV$ by lemma 10.7.1(3), which gives exactly (10.7.7).
$\square$
Pome
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