8.9 Bounded potentials and recursive Hessian currents376A2B30178D

Proof. Choose smooth strictly admissible $u_ j\downarrow u$ by 8.6.6, and put $\sigma _ j=\omega +\mathrm{dd}^{\mathrm{c}}u_ j$. After discarding finitely many terms, $\| u_ j\| _\infty \leq \| u\| _\infty +1$. Indeed, the increasing open sets $\{ u_ j<\sup _ Xu+1\} $ cover $X$, and compactness gives a single index whose open set is all of $X$; the lower bound follows from $u_ j\geq u$.

The closed $m$-cone lies in the closed $2$-cone. In an $\omega $-unitary frame, its eigenvalues satisfy

Thus every coefficient of $\sigma _ j$ is bounded by its trace. The degree-one Chern–Levine–Nirenberg estimate in 8.7.1, applied to these smooth functions, bounds the integral of that trace on $K$ by $C(1+\| u\| _\infty )$. Subtracting the fixed smooth $\omega $ gives the same bound for the coefficient variations of $\mathrm{dd}^{\mathrm{c}}u_ j$. Bounded convergence gives $u_ j\to u$ in local $L^1$, and therefore $\mathrm{dd}^{\mathrm{c}}u_ j\to \mathrm{dd}^{\mathrm{c}}u$ in distributions. The uniform coefficient bounds show that this limit has measure coefficients with the asserted bound. $\square$

Proof. The case $p=1$ is the preceding lemma. Suppose the assertions hold through degree $p-1<m$. Since $u_ p$ is bounded Borel and $D_{p-1}$ has measure coefficients, their product is well defined. Applying $\mathrm{dd}^{\mathrm{c}}$ gives a real current that is both $\mathrm d$- and $\mathrm d^{\mathrm c}$-closed.

For a fixed smooth complementary test form $\eta $, dominated convergence against the fixed signed measures gives

With $j_ p$ fixed, the inductive iterated limit for $D_{p-1}$ can be paired with the smooth form $u_ p^{j_ p}\mathrm{dd}^{\mathrm{c}}\eta $. Integration by parts for the smooth closed factors identifies the resulting pairing with the smooth product on the right of (8.9.4). This proves that identity also when $p=m$.

For $p<m$, expand the smooth product using $\mathrm{dd}^{\mathrm{c}}u_ a^{j_ a}=\sigma _ a^{j_ a}-\omega $. A term with $q$ admissible factors, $0\leq q\leq p$, has the form

Every such $q$ is strictly less than $m$. Apply 8.2.7 to each coefficient test, and then the smooth mass estimate from 8.7.1 to the potential sum. The term $q=0$ is the fixed smooth form $\omega ^ p$. For $q\geq 1$, the sum of its admissible forms is $q(\omega +\mathrm{dd}^{\mathrm{c}}\bar u)$, where $\bar u$ is their average and $\| \bar u\| _\infty \leq M+1$. The result is a coefficient variation bound $C(1+M)^ p$, after choosing the approximations with norm at most $M+1$ as in the preceding proof. For arbitrary uniformly bounded approximations the same argument uses their common bound. The bounds survive each distributional limit, proving that $D_ p$ has order zero and the stated estimate. This completes the induction. No coefficient bound for the unwedged degree-$m$ current is used or asserted. $\square$

Proof. Expand the smooth shifted product over its subsets. Applying (8.9.4) to each subset, in its inherited ordering, gives (8.9.5). The absent indices have no effect on a subset's limit.

Each smooth product is positive: pairing with a decomposable strongly positive complementary test appends $m-p$ nonnegative rank-one $(1,1)$-forms, and polarized Gårding positivity applies. The smooth mixed mass bound is uniform in all indices. Distributional limits preserve positivity and the local trace bound at every step. Positive currents have order zero, so this gives the assertions for $Q_ p$, including the top-degree measure.

The recursion uses only bounded multiplication of canonical values and local distributional differentiation. It is therefore local. Its definition contains no approximation choices, so the proved limit is independent of those choices while the ordering is fixed. For continuous potentials, the approximations converge uniformly by Dini's theorem. The telescoping estimate in 8.7.1 is uniform in all remaining indices and shows that their iterated limit is the same continuous mixed product. This also proves permutation invariance in the continuous case. $\square$

Proof. Every $D_ I$ is both $\mathrm d$- and $\mathrm d^{\mathrm c}$-closed. Differentiating (8.9.5) therefore gives

with zero terms omitted. The coefficient $p-|I|$ counts the omitted indices outside $I$, which proves the first formula; the second is identical with $\mathrm d^{\mathrm c}$ in place of $\mathrm d$. Apply $\mathrm d$ to the second formula. Each pair of omitted indices occurs twice, and $-\mathrm d^{\mathrm c}\omega \wedge \mathrm d\omega =\mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega $ because both forms have odd degree three. This gives the displayed $\mathrm{dd}^{\mathrm{c}}$ formula, with every background derivative retained.

If all but one potential are smooth, applying $\mathrm{dd}^{\mathrm{c}}$ to the bounded potential times a smooth closed product gives its $\mathrm{dd}^{\mathrm{c}}$ wedged with that product. At later smooth steps, closedness again makes the recursion ordinary smooth multiplication. This is independent of the position of the bounded potential, and the subset expansion gives the last assertion. $\square$

Proof. Take $\rho =C|z|^2$ in the chart. On the fixed compact set $\overline U$, a sufficiently large $C$ gives $\mathrm{dd}^{\mathrm{c}}\rho \geq 2\omega $. An upper test $q$ for $u+\rho $ at $x$ gives the upper test $q-\rho $ for $u$ there. Hence

since the second summand is positive and the cone is preserved by adding nonnegative forms. Smooth addition preserves the canonical representative. The same identity applies pointwise to every smooth admissible approximation. Adding a fixed smooth function preserves pointwise decrease and convergence, and its bounded norm on $\overline U$ preserves the common local bound. $\square$

Proof. The eigenvalues of the two perturbed forms relative to $\beta $ are $(\tfrac 12,\tfrac 12,\tfrac 12)$ and $(\tfrac 15,1,1)$. For a diagonal form with relative eigenvalues $\lambda _1,\lambda _2, \lambda _3$, its second Hessian is $(\lambda _1\lambda _2+\lambda _1\lambda _3+\lambda _2\lambda _3)\beta ^3/3$. Consequently the shifted measures of $u,v$ are $\tfrac 14\beta ^3$ and $\tfrac 7{15}\beta ^3$, while their unshifted expressions are $\tfrac 14\beta ^3$ and $0$. Their unshifted traces are negative, so neither original function belongs to the local unshifted class. Adding $\sum _ j|z_ j|^2$ gives an exact correction in this constant closed background. An exact correction $\mathrm{dd}^{\mathrm{c}}\rho =\omega $ is impossible where $\mathrm d\omega \ne 0$. $\square$