12.17.1 The bounded cubic identityEDD3B0732B25

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 $\int _ Xg^3$. $\square$

Proof. We first check that the determinant measure in (12.7.1) does not depend on the Hermitian form used to express it. Let $T$ be a positive $(2,2)$-current, and let $\sigma _ h=h\wedge T$ and $P=P_ h$ be its trace measure and coefficient matrix relative to $h$, as in section 12.7. In an $h$-unitary frame let $G$ be the matrix of $g$, and set

Comparing the defining trace pairings shows that $\sigma _ g=g\wedge T$ and $P_ g$ are the trace measure and coefficient matrix of $T$ relative to $g$, with $\operatorname {tr}P_ g=3$. Since $G>0$ and $\operatorname {tr}P=3$, one has $a>0$ even where $P$ is singular. Moreover $\det P_ g=a^{-3}\det G\det P$ and $g^3=\det G\, h^3$, so the factors $a$ and $\det G$ cancel in $\Gamma _0(T)$. The calculation applies to the matrix density on the whole trace measure, including its singular part.

For $T\in \mathscr G_ h$ put $m=\int _ Xg\wedge T$. Then $0<m\leq c^{-1}$ and $U=T/m\in \mathscr G_ g$. By homogeneity, $k_ T=m k_ U$. Applying theorem 12.9.2 to $g$ gives the norm bound, and applying it again to $h$ gives the lower-mass bound. The supremum here is taken over all of $\mathscr G_ h$, including singular and degenerate currents. $\square$