Lemma 10.2.9410DB42E1141

Proof. For smooth real functions $f,g$ put $B(f,g)=\int _ X\mathrm{d}f\wedge \mathrm{d}^{\mathrm c}g\wedge A$. Only the $(1,1)$-part of $\mathrm df\wedge \mathrm d^{\mathrm c}g$ contributes, and this part equals $\frac{\mathrm i}{4\pi }(\partial f\wedge \bar\partial g+\partial g\wedge \bar\partial f)$, which is symmetric in $f$ and $g$; in particular $\mathrm{d}f\wedge \mathrm{d}^{\mathrm c}f$ has $(1,1)$-part $\frac{\mathrm i}{2\pi }\partial f\wedge \bar\partial f$. Thus $B$ is symmetric. Since $A$ is closed, Stokes' formula (proposition 5.2.6) applied to $\mathrm d(g\, \mathrm d^{\mathrm c}f\wedge A)$ gives

On the compact manifold there are constants $0<c\leq C$ with $c\, \omega \leq \omega _ j\leq C\omega $ for all $j$, by lemma 6.14.1. For $(1,1)$-forms $\gamma _ j,\gamma '_ j$ we have the telescoping identity

Applying it with $\gamma _ j=\omega _ j$, $\gamma '_ j=c\, \omega $ and with $\gamma _ j=C\omega $, $\gamma '_ j=\omega _ j$, we see that the forms $A-c^{n-1}\omega ^{n-1}$ and $C^{n-1}\omega ^{n-1}-A$ are sums of products of semipositive $(1,1)$-forms. Such products, and their products with the semipositive form $\mathrm i\partial f\wedge \bar\partial f$, are strongly positive (proposition 5.3.2). Hence $B(f,f)$ lies between $c^{n-1}$ and $C^{n-1}$ times $\int _ X\frac{\mathrm i}{2\pi }\partial f\wedge \bar\partial f\wedge \omega ^{n-1}$. At a point, if $\theta _1,\ldots ,\theta _ n$ is a unitary coframe, that is, $\omega =\mathrm i\sum _ j\theta _ j\wedge \bar\theta _ j$, and $\partial f=\sum _ ja_ j\theta _ j$, then $\mathrm i\partial f\wedge \bar\partial f\wedge \omega ^{n-1}/(n-1)!=\sum _ j|a_ j|^2\omega ^ n/n!$ and $|\mathrm df|^2=2\sum _ j|a_ j|^2$; so the last integral is $\frac{(n-1)!}{4\pi }\| \mathrm df\| _{L^2}^2$. By lemma 10.2.5(4), $B$ is therefore an equivalent inner product on the closed subspace of real functions of mean zero in $H^1$.

The sesquilinear extension $B_{\mathbb C}(f_1+\mathrm if_2,g_1+\mathrm ig_2)=B(f_1,g_1)+B(f_2,g_2)+\mathrm i\bigl (B(f_2,g_1)-B(f_1,g_2)\bigr )$ to complex-valued functions is, by the symmetry of $B$, a Hermitian form with $B_{\mathbb C}(f,f)=B(f_1,f_1)+B(f_2,f_2)$, hence an equivalent inner product on the complex Hilbert space of functions of mean zero in $H^1$. The Riesz representation theorem (proposition 6.2.1), applied to the bounded conjugate-linear functional $g\mapsto -\int _ X\overline g\, \eta $, gives $F$ in this space with $B_{\mathbb C}(F,g)=-\int _ X\overline g\, \eta $ for all $g$. For real $g$ the right side is real and $B_{\mathbb C}(F,g)=B(\operatorname{Re}F,g)+\mathrm iB(\operatorname{Im}F,g)$, so $f=\operatorname{Re}F$ is a real function of mean zero in $H^1$ with $B(f,g)=-\int _ Xg\eta $ for every real mean-zero $g\in H^1$. Since $\int _ X\eta =0$ and $B(f,1)=0$, the identity holds for all $g\in H^1$; this is the equation $\mathrm{dd}^{\mathrm{c}}f\wedge A=\eta $ in the weak sense.

It remains to prove that $f$ is smooth. In a chart, with $\eta =\tilde\eta \, \mathrm dx$, the weak equation reads $\mathcal A(f,w)=-\int \tilde\eta \, \overline w\, \mathrm dx$ for $w\in C^\infty _ c$, where $\mathcal A(f,w)=B(f,\overline w)=\int \sum _{i,j}a^{ij}\partial _ jf\, \overline{\partial _ iw}\, \mathrm dx$ and the real symmetric matrix $(a^{ij})$ is the density of the quadratic form $\mathrm df\wedge \mathrm d^{\mathrm c}f\wedge A$. By the bounds just proved, $(a^{ij})$ is uniformly positive definite on compact subsets of the chart. This is a form of the kind considered in lemma 10.2.4, with $N=1$, and since $\tilde\eta $ is smooth, that lemma, applied with the cutoffs $\chi _ k$ on relatively compact open subsets of the charts, shows that every $\chi _ kf$ has square integrable distributional derivatives of all orders. Hence $f$ lies in every $H^ r$ by lemma 10.2.1(2), and it is smooth by lemma 10.2.1(3). If $f$ and $f'$ are two solutions, then $B(f-f',f-f')=0$, so $f-f'$ is constant on the connected manifold $X$, and it vanishes since it has mean zero. $\square$