We formulate the result for a connected compact complex manifold $Y$ of dimension $n\geq 1$ with a Hermitian form $h$, since it will be applied with different metrics in section 23.1. Consider smooth functions
\[ v_ j=s_ ju_ j,\qquad u_ j\in \mathrm{PSH}(Y,h),\qquad \sup _ Yu_ j=0,\qquad 0<s_ j\longrightarrow 0. \]
Suppose that their Monge–Ampère masses $\int _ Y(h+\mathrm{dd}^{\mathrm{c}}v_ j)^ n$ equal a fixed number $b$ and that
\[ (h+\mathrm{dd}^{\mathrm{c}}v_ j)^ n\rightharpoonup h^ n+\sigma . \]
The lower bound $h+\mathrm{dd}^{\mathrm{c}}v_ j\geq (1-s_ j)h$ shows that $\sigma $ is positive. Although $\mathrm{dd}^{\mathrm{c}}v_ j\to 0$ in the trace norm, its intermediate powers can have nonzero limits, and these limits determine $\sigma $.
Proposition 13.15.20.708664A6CF93 After passage to a common subsequence, the positive forms
\[ H_ j=\mathrm{dd}^{\mathrm{c}}v_ j+s_ jh=s_ j(h+\mathrm{dd}^{\mathrm{c}}u_ j) \]
satisfy $H_ j^ k\rightharpoonup K_ k$ for $1\leq k\leq n$. The currents $K_ k$ are positive and closed, and $K_1=K_ n=0$. For every smooth real $(n-k,n-k)$-form $\eta $ with $\mathrm{d}\eta =0$ or $\mathrm{dd}^{\mathrm{c}}\eta =0$,
\[ \int _ Y K_ k\wedge \eta =0. \]
Moreover,
13C111AE3D65
13.15.21
\begin{equation} \sigma =\sum _{k=2}^{n-1}\binom nk h^{n-k}\wedge K_ k. \end{equation}
If $\vartheta $ is a smooth closed semipositive $(1,1)$-form on $Y$, then $\sigma $ is supported on $\{ \vartheta ^ n=0\} $. In dimension three the same conclusion holds for every smooth semipositive $(1,1)$-form $\vartheta $ with $\mathrm{dd}^{\mathrm{c}}\vartheta =0$.
Proof.
For $s_ j<1/2$, positivity in the expansion
\[ (h+\mathrm{dd}^{\mathrm{c}}v_ j)^ n =\sum _{k=0}^ n\binom nk(1-s_ j)^{n-k}h^{n-k}\wedge H_ j^ k \]
gives a uniform bound for each $\int _ Yh^{n-k}\wedge H_ j^ k$. In local unitary coordinates, the coefficient matrix of the smooth positive form $H_ j^ k$ on exterior powers is positive semidefinite, so its entries are bounded in absolute value by its trace, up to a dimensional constant. The trace bounds therefore bound every coefficient in total variation, and weak compactness of measures gives a common subsequence and positive limits $K_ k$.
For $n\geq 2$, choose a Gauduchon form $\gamma $. By proposition 11.5.1,
\[ \int _ Y H_ j\wedge \gamma ^{n-1} =s_ j\int _ Yh\wedge \gamma ^{n-1}. \]
Since $\gamma $ and $h$ are equivalent smooth metrics, it follows that $H_ j\to 0$ in coefficient total variation. For $n=1$ the same conclusion follows from $\int _ YH_ j=s_ j\int _ Yh$. Thus $K_1=0$ in all dimensions. Since $\mathrm{d}H_ j=s_ j\mathrm{d}h$,
\[ \mathrm{d}(H_ j^ k)=ks_ j\mathrm{d}h\wedge H_ j^{k-1}\longrightarrow 0 \]
as currents, because the preceding trace bounds control multiplication by the fixed smooth form $\mathrm{d}h$. Hence every $K_ k$ is closed.
The smooth form $(\mathrm{dd}^{\mathrm{c}}v_ j)^ k$ is both $\mathrm{d}$-exact and $\mathrm{dd}^{\mathrm{c}}$-exact:
\[ (\mathrm{dd}^{\mathrm{c}}v_ j)^ k =\mathrm{d}\bigl (\mathrm d^{\mathrm c}v_ j\wedge (\mathrm{dd}^{\mathrm{c}}v_ j)^{k-1}\bigr ) =\mathrm{dd}^{\mathrm{c}}\bigl (v_ j(\mathrm{dd}^{\mathrm{c}}v_ j)^{k-1}\bigr ). \]
Its integral against either kind of test form $\eta $ in the statement is therefore zero by Stokes' formula. Expanding $\mathrm{dd}^{\mathrm{c}}v_ j=H_ j-s_ jh$ gives
\[ \int _ Y\eta \wedge H_ j^ k =-\sum _{r=0}^{k-1}\binom kr(-s_ j)^{k-r} \int _ Y\eta \wedge h^{k-r}\wedge H_ j^ r =O(s_ j). \]
Each integral on the right is bounded by the trace bound for $H_ j^ r$. Taking $k=n$ and $\eta =1$ shows that the positive measures $H_ j^ n$ tend to zero in total variation, which proves $K_ n=0$; the other vanishing integrals follow by weak convergence. Finally, letting $j\to \infty $ in the positive expansion above, the terms with $k=0$ tend to $h^ n$ and those with $k=1$ and $k=n$ vanish in the limit, which proves (13.15.21).
For closed semipositive $\vartheta $, apply the vanishing integrals to $\eta =\vartheta ^{n-k}$. The measure $K_ k\wedge \vartheta ^{n-k}$ is positive and has zero mass. On every relatively compact open set where $\vartheta >0$, it dominates a positive multiple of the trace measure of $K_ k$, so $K_ k$ vanishes there. The decomposition gives the asserted support. In dimension three only $K_2$ occurs, and the same argument applies to the $\mathrm{dd}^{\mathrm{c}}$-closed test form $\vartheta $ itself.
$\square$
In dimensions at most two the decomposition forces $\sigma =0$. A threefold carrying a pluriclosed Hermitian metric, that is, a Hermitian form $\vartheta $ with $\mathrm{dd}^{\mathrm{c}}\vartheta =0$, has no nonzero defect $\sigma $ of this form either. In higher dimensions, the vanishing integrals impose necessary conditions on each intermediate current; they do not by themselves produce a smooth sequence with prescribed intermediate limits.
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