Lemma 12.17.23BBBC535BFA2

Proof. Locally write $u=p-a$, with $p$ bounded plurisubharmonic and $a$ smooth. Expanding in $p$ and $a$ and using multilinearity, we obtain closed currents $C_ r=(\mathrm{dd}^{\mathrm{c}}u)^ r$, not necessarily positive, with $C_0=1$. The recursive definition of bounded products, expanded in the same way, gives

By lemma 5.27.2 and invariance under smooth corrections, the weighted product $u\mathrm{dd}^{\mathrm{c}}u$ does not depend on the local decomposition. On a smaller ball choose smooth bounded plurisubharmonic $p_ j$ decreasing to $p$, and put $u_ j=p_ j-a$. The same lemma gives $u_ j\mathrm{dd}^{\mathrm{c}}u_ j\to u\mathrm{dd}^{\mathrm{c}}u$, while boundedness gives $u_ j^2\to u^2$ in $L^1$. Hence

The smooth currents on the left are positive, so their weak limit $E_ u$ is positive and has order zero.

Now expand the cube. Repeated use of distributional Stokes gives

Since $\mathrm{dd}^{\mathrm{c}}g$ is closed, $\int _ X\mathrm{dd}^{\mathrm{c}}(u^2)\wedge \mathrm{dd}^{\mathrm{c}}g=0$, so the last integral equals $-\int _ XE_ u\wedge \mathrm{dd}^{\mathrm{c}}g$. This proves (12.17.3). If $-\mathrm{dd}^{\mathrm{c}}g\geq 0$, the correction term is nonnegative, and $u=0$ attains the lower bound $\int _ Xg^3$. $\square$