Proposition 9.11.69FEAE8B85485E

Proof. Convolve $U$, $V$ and $k$ with the same nonnegative radial kernel $\varrho _ j$ on the torus, of radius $r_ j\downarrow 0$, and write

Translations preserve positivity, so $A_ j\geq 0$ and $B_ j\geq \beta K_0$ with $\beta =1/4$. The forms $k_ j$ converge smoothly, and $V_ j$ converges in $C^2$. On lifted coordinate balls, choose a common positive quadratic function $q$ with $\mathrm{dd}^{\mathrm{c}}q\geq k$. Convolution gives $q*\varrho _ j=q+b r_ j^2$ for a fixed $b>0$. The radial averages of the bounded plurisubharmonic functions $U+q$ and $V+q$ decrease as the radii decrease. Hence $U_ j+b r_ j^2+q$ and $V_ j+b r_ j^2+q$ decrease to $U+q$ and $V+q$; the identical constants cancel in the difference.

Put $\phi _ j=U_ j-V_ j$ and $T_ j=\sum _{r=0}^{n-1}A_ j^ r\wedge B_ j^{n-1-r}$. The identities $\mathrm{dd}^{\mathrm{c}}k_ j=0$ and $\mathrm dk_ j\wedge \mathrm d^{\mathrm c}k_ j=0$ persist under convolution, since the torsion is still pulled back from $E_1\times E_2$. Differentiating the monomials therefore gives $\mathrm{dd}^{\mathrm{c}}T_ j=0$. Also $T_ j\geq B_ j^{n-1}\geq \beta ^{n-1}K_0^{n-1}$. Stokes' theorem applied to $\mathrm{dd}^{\mathrm{c}}(\phi _ j^2)$ yields

Only the $\mathrm{dd}^{\mathrm{c}}$-closedness of $T_ j$ is needed here.

We treat the four terms on the left separately. Expand $A_ j^ n$ with respect to the local plurisubharmonic potential $U_ j+b r_ j^2+q$ and the smoothly converging coefficient $k_ j-\mathrm{dd}^{\mathrm{c}}q$. The weighted limit lemma 5.27.2 gives $\int U_ jA_ j^ n\to \int U(k+\mathrm{dd}^{\mathrm{c}}U)^ n$. Decreasing convergence and the uniform convergence of $V_ j$ give the corresponding limit for $\int V_ jA_ j^ n$. Since $B_ j^ n$ has uniformly converging continuous densities, bounded pointwise convergence and dominated convergence give the limit of $\int U_ jB_ j^ n$. The term $\int V_ jB_ j^ n$ converges by $C^2$ convergence. A finite partition of unity gives the global statements. Since the two limiting measures are equal, the left-hand side of (9.11.70) tends to zero.

The flat Fourier Poincaré inequality now shows that the $L^2$ norm of $\phi _ j$ minus its average tends to zero. Bounded convergence gives $\phi _ j\to U-V$ in $L^2$, so $U-V$ is constant almost everywhere. Local smooth corrections and the canonical plurisubharmonic representatives extend this equality to every point. $\square$