Lemma 11.22.340BA141971D9

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. $\square$