11.22 Pluriharmonic currentsB57478F1D4E4

Proof. Use local coordinates $(z,w_1,\ldots ,w_{m-1})$ with $D=\{ z=0\} $. Both $R$ and $\mathrm{dd}^{\mathrm{c}}R$ have measure coefficients supported on $D$. Suppose first that $q<m$. For every compactly supported smooth form $\eta $ of bidegree $(m-q-1,m-q-1)$,

where the first equality holds because $\mathrm{dd}^{\mathrm{c}}R$ is a measure supported on $D$, on which $|z|^2$ vanishes, and the last one because the terms containing $|z|^2$ or its first derivatives vanish on $D$. Thus $R\wedge \mathrm{i}\, \mathrm{d}z\wedge \mathrm{d}\bar z=0$, which removes the coefficients of $R$ containing neither normal differential.

Fix a constant positive Hermitian form $\tau $ in the $w$ variables. The current $M_\tau =R\wedge \tau ^{m-q-1}$ is positive of bidegree $(m-1,m-1)$, and $M_\tau \wedge \mathrm{i}\, \mathrm{d}z\wedge \mathrm{d}\bar z=0$. Its Hermitian matrix of measure densities therefore has zero diagonal entry in the normal direction, and positivity removes the corresponding row and column, so

Put $s=m-q-1$. The powers $\tau ^ s$, as $\tau $ varies over the positive Hermitian forms, span the real horizontal $(s,s)$-forms. Indeed, a linear functional vanishing on these powers vanishes on a polynomial over an open cone; polarization makes it vanish on products of arbitrary real $(1,1)$-forms, and these products span the forms of that bidegree. Their products with a horizontal one-form span the complementary test forms for the mixed coefficients of $R$. The identities above therefore remove every coefficient containing exactly one normal differential. For $q=m$, every top-degree coefficient already contains both normal differentials, and this reduction is unnecessary.

The remaining coefficients are measures on $\{ z=0\} $ multiplying both normal differentials. They therefore define a unique current $S$ on $D$ with $R=j_*S$. Testing against extensions of strongly positive forms on $D$ proves positivity of $S$, and uniqueness makes these local currents agree on overlaps. Pushforward commutes with $\mathrm{dd}^{\mathrm{c}}$ and is injective for a closed embedding, so $\mathrm{dd}^{\mathrm{c}}R=0$ implies $\mathrm{dd}^{\mathrm{c}}S=0$.

When $q=1$, the current $S$ is a nonnegative distribution on $D$. The equation $\mathrm{dd}^{\mathrm{c}}S=0$ and Weyl's lemma make it a smooth nonnegative pluriharmonic function. If $D$ is compact and connected, the maximum principle makes it a constant $c\geq 0$, proving the last assertion. $\square$

Proof. The positive closed current $\Gamma =F_*(\Omega ^{N-p})$ has bidegree $(n-p,n-p)$ on $X$. The classical $\mathrm{dd}^{\mathrm{c}}$-lemma for currents on compact Kähler manifolds gives a smooth real closed form $\gamma $ of this bidegree and a current $U$ with $\Gamma =\gamma +\mathrm{dd}^{\mathrm{c}}U$. For $p=n$, a closed current of degree zero is already a smooth locally constant function, and one may take $U=0$. Since $S$ is smooth, pairing it with $U$ is legitimate, and

The pairing with the fixed smooth form $\gamma $ is bounded in absolute value by a constant times the trace mass $\int _ XS\wedge \omega ^{n-p}$ of the positive form $S$. This proves the estimate. $\square$

Proof. Let $\Delta $ be the real Hodge Laplacian of the Kähler metric. Its heat operators $H_ t=\mathrm e^{-t\Delta }$ are smoothing for $t>0$. The Kähler identities give $\Delta =2\Delta _\partial =2\Delta _{\bar\partial }$, so $H_ t$ preserves bidegree, reality and the equation $\mathrm{dd}^{\mathrm{c}}R=0$.

We recall why $H_ tR\to R$ uniformly for a continuous form $R$. The Weitzenböck formula writes $\Delta =\nabla ^*\nabla +A$, with a bounded curvature term $A$ of order zero. For smooth initial data $\beta $, the heat equation and the maximum principle give

where $C$ depends only on the fixed metric and the degree. Indeed, the scalar equation for $|H_ t\beta |^2$ has a nonpositive gradient term and a curvature term bounded above by $2C|H_ t\beta |^2$. The classical smooth heat solution converges uniformly to its smooth initial data. Approximating a continuous form by smooth forms and using this bound proves uniform convergence for continuous initial data. The resulting heat solution is smooth for $t>0$, and since the commutation with $\mathrm{dd}^{\mathrm{c}}$ extends to currents, $\mathrm{dd}^{\mathrm{c}}H_ tR=0$.

Set $\epsilon _ t=\| H_ tR-R\| _{C^0}$, which tends to zero. A fixed constant $c>0$ satisfies

for every continuous real $(p,p)$-form $E$; to see this, evaluate on decomposable unit $p$-vectors, on which $\omega ^ p$ has a uniformly positive value. Consequently

is smooth and positive. Since $\omega $ is Kähler, $\mathrm{dd}^{\mathrm{c}}\omega ^ p=0$, and hence $\mathrm{dd}^{\mathrm{c}}R_ t=0$. Moreover, $R_ t\to R$ uniformly. The Hodge heat flow also commutes with $\mathrm{d}$, and $\omega ^ p$ is closed, so $\mathrm{d}R=0$ implies $\mathrm{d}R_ t=0$. Choosing any sequence $t_ j\downarrow 0$ proves the assertion. $\square$

Proof. Fix a local trivialization $\pi ^{-1}(U)=U\times F$ of the bundle, with base coordinates $w=(w_1,\ldots ,w_\ell )$ and a local fiber coordinate $z$. The coefficient measures of $S$ have a positive Hermitian matrix of densities relative to their trace measure. The identity $S\wedge \pi ^*\eta ^\ell =0$ makes the vertical diagonal entry zero, and the inequality $|s_{\alpha \bar z}|^2\leq s_{\alpha \bar\alpha }s_{z\bar z}$ then removes all mixed entries. Thus

with distribution coefficients. In the equation $\partial \bar\partial S=0$, the coefficient containing both vertical differentials and the horizontal pair $(\alpha ,\beta )$ gives $\partial _ z\partial _{\bar z}s_{\alpha \bar\beta }=0$.

Fix $v\in \mathbb C^\ell $ and a nonnegative smooth function $\xi $ with compact support in $U$. Pairing the positive coefficient distribution $\sum _{\alpha ,\beta }s_{\alpha \bar\beta }v_\alpha \bar v_\beta $, in the base variables, with $\xi $ times the Euclidean volume form defines a nonnegative harmonic distribution on the whole fiber $F$. Its definition does not depend on the fiber coordinate, since the remaining differentials are horizontal. Weyl's lemma makes it a nonnegative harmonic function. Such a function is constant on $\mathbb C$ by the Liouville theorem for harmonic functions, and on a compact connected curve by the maximum principle.

Polarization in $v$ and linearity in the base test function show that every coefficient $s_{\alpha \bar\beta }$ is the pullback of a distribution $r_{\alpha \bar\beta }$ on $U$. To identify this distribution, pair against any nonnegative smooth fiber density of total integral one; this also proves positivity of the resulting current $R_ U$ on $U$ and injectivity of the pullback. Consequently $\mathrm{dd}^{\mathrm{c}}S=\pi ^*(\mathrm{dd}^{\mathrm{c}}R_ U)=0$ implies $\mathrm{dd}^{\mathrm{c}}R_ U=0$. On overlaps, injectivity makes the currents $R_ U$ agree, so they glue to a current $R$, and injectivity also proves uniqueness. If $\mathrm{d}S=0$, the same argument gives $\mathrm{d}R=0$. $\square$