Lemma 10.7.295269D214AF0

Proof. If $e'=ge$ is a second holomorphic frame, with $g$ holomorphic and nowhere zero, then its weight is $p'=p-\log |g|^2$, and for a form $u'$ we have $\partial (gu')-\partial p\wedge gu'=g(\partial u'-\partial p'\wedge u')$, since $\partial \log |g|^2=g^{-1}\partial g$; so $D'$ does not depend on the frame. In a frame we have $D'D''+D''D'=\partial \bar\partial p\wedge $, and $\mathrm i\partial \bar\partial p=2\pi c_1(F,h)$. We prove the Kähler identities at a point, in coordinates and a frame adapted to $\sigma $ and $h$, and deduce the Bochner–Kodaira formula from them.

Fix a point, choose holomorphic coordinates centered at it in which $\sigma =\mathrm i\sum _ j\mathrm dz_ j\wedge \mathrm d\bar z_ j$ at the point and the first derivatives of the coefficients of $\sigma $ vanish there (lemma 10.2.11, after replacing the coordinates $z$ by $z/\sqrt2$), and a frame with $\partial p=0$ at the point: replacing a frame $e$ by $e\exp (\sum _ j\partial _ jp(0)z_ j)$ replaces $p$ by $p-2\operatorname{Re}\sum _ j\partial _ jp(0)z_ j$. Let $\epsilon _ k,\bar\epsilon _ k$ denote exterior multiplication by $\mathrm dz_ k,\mathrm d\bar z_ k$, and $\iota _ k,\bar\iota _ k$ their pointwise adjoints, the contractions. At the point they satisfy $\iota _ j\epsilon _ k+\epsilon _ k\iota _ j=\delta _{jk}$ and $\bar\iota _ j\bar\epsilon _ k+\bar\epsilon _ k\bar\iota _ j=\delta _{jk}$, while operators of different types anticommute; moreover $\Lambda _\sigma =-\mathrm i\sum _ j\bar\iota _ j\iota _ j$, as the adjoint of $\mathrm i\sum _ j\epsilon _ j\bar\epsilon _ j$.

The formal adjoints of $\sum _ k\epsilon _ k\partial _ k$ and $\sum _ k\bar\epsilon _ k\partial _{\bar k}$ are $-\sum _ k\iota _ k\partial _{\bar k}$ and $-\sum _ k\bar\iota _ k\partial _ k$, with $\iota _ k,\bar\iota _ k$ computed for the metric at each point, plus zero-order terms whose coefficients are linear in the first derivatives of the coefficients of $\sigma $ and of $\mathrm e^{-p}$. Indeed, in the frame $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ the operators $\epsilon _ k$ have constant matrices, and the pointwise inner product of $F$-valued forms times the density of $\mathrm dV_\sigma $ is $\overline w^{\mathsf T}Hu$ for a smooth positive definite Hermitian matrix $H$ whose entries are smooth functions of the coefficients of $\sigma $ times $\mathrm e^{-p}$. For a constant matrix $A$ and compactly supported smooth $u,w$, integration by parts and $\partial _ k\overline w=\overline{\partial _{\bar k}w}$ give $\int \overline w^{\mathsf T}HA\, \partial _ ku=-\int \overline{(\partial _{\bar k}w)}^{\mathsf T}HAu-\int \overline w^{\mathsf T}(\partial _ kH)Au$, so the formal adjoint of $A\partial _ k$ is $-A^\dagger \partial _{\bar k}-H^{-1}A^*(\partial _{\bar k}H)$, where $A^\dagger =H^{-1}A^*H$ is the pointwise adjoint of $A$ and $A^*$ the conjugate transpose; the same computation applies with $\partial _ k$ and $\partial _{\bar k}$ exchanged. Since these first derivatives, and $\partial p$, vanish at the point, there

where $\partial _ k,\partial _{\bar k}$ act on the coefficients. The anticommutation rules give $\bar\iota _ j\iota _ j\epsilon _ k-\epsilon _ k\bar\iota _ j\iota _ j=\delta _{jk}\bar\iota _ j$ and $\bar\iota _ j\iota _ j\bar\epsilon _ k-\bar\epsilon _ k\bar\iota _ j\iota _ j=-\delta _{jk}\iota _ j$, hence $[\Lambda _\sigma ,D']=-\mathrm i\sum _ k\bar\iota _ k\partial _ k$ and $[\Lambda _\sigma ,D'']=\mathrm i\sum _ k\iota _ k\partial _{\bar k}$ at the point, since the coefficients of $\Lambda _\sigma $ in the frame $\mathrm dz_ I\wedge \mathrm d\bar z_ J$ have vanishing first derivatives there. These are the Kähler identities

which therefore hold everywhere, the point being arbitrary. Substituting them into $\Delta ''=D''D''{}^*+D''{}^*D''$ and $\Delta '=D'D'{}^*+D'{}^*D'$, the mixed compositions $D''\Lambda _\sigma D'$ and $D'\Lambda _\sigma D''$ cancel in the difference, and the curvature identity gives the Bochner–Kodaira formula

that is, $\Delta ''-\Delta '=B_\sigma $. For a compactly supported smooth $(n,1)$-form $v$, the form $D'v$ vanishes for degree reasons, and integrating $\langle (\Delta ''-\Delta ')v,v\rangle _\sigma $ over $M$, using the definition of the formal adjoints, gives (10.7.3). $\square$