Lemma 10.2.12C7E6E1462EE1

Proof. Both sides of each identity are differential operators, or bundle maps, with smooth coefficients, acting on distributions and currents as explained at the beginning of this section. Hence it suffices to verify the identities for smooth $f$ and $T$, at an arbitrary point. By lemma 10.2.11, there are holomorphic coordinates centered at the point in which $\omega =\frac{\mathrm i}{2}\sum _ j\mathrm dz_ j\wedge \mathrm d\bar z_ j$ at the point and the first derivatives of the coefficients of $\omega $, hence of the metric coefficients, vanish there. Thus (10.2.7) gives $Lf=-\sum _ j(\partial _{x_ j}^2+\partial _{y_ j}^2)f=-4\sum _ j\partial _ j\partial _{\bar j}f$ there, while $\theta _ j=\mathrm dz_ j/\sqrt2$ is a unitary coframe and

Hence $\Lambda \mathrm{dd}^{\mathrm{c}}f=\pi ^{-1}\sum _ jf_{j\bar j}=-Lf/(4\pi )$. For the second identity, diagonalize the Hermitian matrix of $T$ by a unitary change of coframe; by linearity it suffices to treat $T=\mathrm i\theta _1\wedge \bar\theta _1$. Write $\sqrt2\theta _ j=e_{2j-1}+\mathrm ie_{2j}$ with real covectors $e_ a$; they form an orthonormal basis, positively oriented since $\omega ^ n/n!=e_1\wedge \cdots \wedge e_{2n}$, and $\omega _ j=\mathrm i\theta _ j\wedge \bar\theta _ j=e_{2j-1}\wedge e_{2j}$. By the explicit formula for $*$, the star of $\omega _1$ is $\omega _2\wedge \cdots \wedge \omega _ n$. On the other hand $\omega ^{n-1}/(n-1)!$ is the sum of all products of $n-1$ distinct $\omega _ j$, while $\omega _1\wedge \omega ^{n-2}/(n-2)!$ is the sum of those products that contain $\omega _1$; their difference is exactly $\omega _2\wedge \cdots \wedge \omega _ n$, and $\Lambda \omega _1=1$.

If $\mathrm dT=0$ and $\Lambda T$ is constant, the second identity and $\mathrm d\omega =0$ give $\mathrm d*T=0$, hence $\mathrm d^*T=-*\mathrm d*T=0$, and $\mathcal LT=\mathrm d\mathrm d^*T+\mathrm d^*\mathrm dT=0$. In dimension one a $(1,1)$-current is $T=(\Lambda T)\omega $, so $*T=\Lambda T$ is constant and the same conclusion holds. $\square$