Lemma 10.8.1D973E523027E

Proof. The Bochner–Kodaira identity (10.7.3) of lemma 10.7.2 holds for every smooth metric on $F$; we apply it to the metric $ah$. Let $\delta _ h=D''{}^*$ be the formal adjoint for the metric $h$ and the Kähler form $\sigma $, and $C_ a=(\bar\partial a\wedge )^*$ the contraction with the dual of $\bar\partial a$. In a frame, the weight of $ah$ is $p-\log a$, so

On $(n,1)$-forms, the commutator with the last term is $[\frac{\mathrm i}{a}\partial a\wedge \bar\partial a,\Lambda _\sigma ]=a^{-1}C_ a^*C_ a$. Indeed, where $\partial a\neq 0$, choose a $\sigma $-unitary coframe with $\partial a=|\partial a|\theta _1$, so that $\bar\partial a=|\partial a|\bar\theta _1$, and put $e=\theta _1\wedge \cdots \wedge \theta _ n$. By lemma 10.7.1(1), the left side multiplies the coefficient $v_1$ of $e\wedge \bar\theta _1$ by $|\partial a|^2/a$ and annihilates the other coefficients; so does the right side, because $\bar\partial a\wedge fe=(-1)^ n|\partial a|f\, e\wedge \bar\theta _1$, whence $C_ av=(-1)^ n|\partial a|v_1e$. Where $\partial a=0$, both sides vanish. Writing (10.7.3) for $ah$ in terms of $h$, expanding $a|\delta _ hv-a^{-1}C_ av|^2$, cancelling the term $a^{-1}|C_ av|^2$ that appears on both sides, and dropping the nonnegative term $\| D'{}^*v\| ^2$, we obtain for compactly supported smooth $v$, with all integrals taken with respect to $h$, $\sigma $ and $\mathrm dV_\sigma $,

The inequality $2\operatorname{Re}\langle A,B\rangle \geq -b|A|^2-b^{-1}|B|^2$ and the identity $b^{-1}|C_ av|^2=\langle [\frac{\mathrm i}{b}\partial a\wedge \bar\partial a,\Lambda _\sigma ]v,v\rangle $ consequently give

Consider the Hilbert spaces of $(n,0)$-, $(n,1)$- and $(n,2)$-forms with the respective weights $h/(a+b)$, $h$ and $ah$, and the operators $T=D''$ and $S=D''$ with maximal domains between them. Testing against compactly supported forms shows that $T^*=(a+b)\delta _ h$, so $\| T^*v\| ^2$ is the first term on the right of (10.8.3), and $\| Sv\| ^2$ is the second. Compactly supported smooth forms are dense in $\operatorname{Dom}T^*\cap \operatorname{Dom}S$ for the graph norm, by lemma 10.7.4, applied with the metrics $h/(a+b)$, $h$ and $ah$, whose ratios $a+b$ and $a$ are bounded.

The form $g$ lies in $\ker S$, and in the second space by lemma 10.7.1(3), because $\sigma \geq \omega /2$. The operator $[H,\Lambda _\sigma ]$ is positive definite on $(n,1)$-forms by lemma 10.7.1(1). As in the proof of lemma 10.7.6, the pointwise Cauchy–Schwarz inequality for this operator, (10.8.3) and the density of compactly supported smooth forms give the hypothesis of theorem 6.2.4, which yields a solution of $\bar\partial u=g$ whose squared norm in the first space is at most $\int \langle [H,\Lambda _\sigma ]^{-1}g,g\rangle \, \mathrm dV_\sigma $. This integral, and the squared norm of $(n,0)$-forms times the volume form, do not depend on the Hermitian form used, by lemma 10.7.1(3); hence the bound is $M_ g$, and we obtain (10.8.2). $\square$