Lemma 8.13.70F8C3F6B0122

Proof. First estimate the torsion of $T_{p,q}$, writing $r=p+q$. For smooth admissible potentials, the product rule expresses $\mathrm{dd}^{\mathrm{c}}T_{p,q}$ as a sum of fixed smooth metric forms wedged with products of $\sigma _ u,\sigma _ V$ of degrees $r,r-1,r-2$. The terms of degree $r$ come from differentiating the background power; they cannot be discarded. Indeed, $\mathrm d\sigma _ u=\mathrm d\sigma _ V=\mathrm d\omega $, $\mathrm d^{\mathrm c}\sigma _ u=\mathrm d^{\mathrm c}\sigma _ V=\mathrm d^{\mathrm c}\omega $ and $\mathrm{dd}^{\mathrm{c}}\sigma _ u=\mathrm{dd}^{\mathrm{c}}\sigma _ V=\mathrm{dd}^{\mathrm{c}}\omega $. Every term therefore contains either $\mathrm{dd}^{\mathrm{c}}\omega $ or $\mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega $, with its product-rule multiplicity and the appropriate remaining background power. Terms with negative numbers of factors are absent.

All variable degrees are at most $r<m$. Apply 8.2.7 to each term. Expanding the resulting positive power of the sum of its variable factors gives the pointwise bound

For $q=0$ every variable factor is $\sigma _ u$, so the sharper bound is $C(M_{p,0}+M_{p-1,0}+M_{p-2,0})$. The constant bounds fixed metric coefficients on a finite coordinate cover and the finitely many multiplicities; it is independent of the potentials and their bounds.

Choose uniformly bounded smooth decreasing approximations to $u,V$. By 8.12.4, the torsion measures converge weakly; by 8.11.12, all positive measures in these bounds also converge weakly. Against nonnegative continuous tests, lower semicontinuity of total variation preserves both bounds. They are consequently inequalities of measures and may be restricted to the Borel set $E$.

Put $w=\max \{ u,V+b\} $ and $h=w-u$. The mixed maximum theorem 8.12.2, with one moving slot and the fixed slots defining $T_{p,q}$, gives

All products here are the corresponding positive mixed measures. After integration, this implies

Regularized maxima of the smooth decreasing approximations give a uniformly bounded decreasing smooth list $w_ j\downarrow w$. Stokes' theorem on compact $X$ gives for these lists

Joint decreasing convergence passes the positive mixed measures on the left to their limits. The continuous two-variable weight $F(x,y)=y-x$ in 8.12.4 passes the right side to $\int _ Xh\mathrm{dd}^{\mathrm{c}}T_{p,q}$. No monotonicity of $w_ j-u_ j$ is needed. Since $h$ is zero outside $E$ and $0\leq h\leq s$,

The torsion bounds prove both assertions. The argument never passes a varying measure to a weak limit after restriction to $E$. $\square$