Proposition 8.7.1.D2A0F255F625 For continuous admissible $u_1,\ldots ,u_ k$, $1\leq k\leq m$, the current
\[ (\omega +\mathrm{dd}^{\mathrm{c}}u_1)\wedge \cdots \wedge (\omega +\mathrm{dd}^{\mathrm{c}}u_ k) \wedge \omega ^{n-m} \]
is well defined as the limit of uniform smooth admissible approximations. It is positive, local, and continuous when its potentials converge uniformly. Its local masses are bounded in terms of uniform bounds on the potentials. In particular $H_ m(u)$ is a positive Radon measure.
Proof.
8.6.1, 8.6.2 provide the smooth approximations without using any continuous Hessian product or domination input. For a smooth admissible $u$ put $\sigma =\omega +\mathrm{dd}^{\mathrm{c}}u$. Since $\mathrm d\sigma =\mathrm d\omega $, the coefficients of $\mathrm{dd}^{\mathrm{c}}(\chi \sigma ^{r-1}\wedge \omega ^{n-r})$ are sums of smooth test factors paired with powers $\sigma ^ q$, $q\leq r-1$, where $\chi $ is a fixed cutoff. The absolute-form lemma and integration by parts give the Chern–Levine–Nirenberg bound by induction on $r\leq m$:
\[ \int _ K\sigma ^ r\wedge \omega ^{n-r} \leq C_{K,U,\omega }(1+\| u\| _{L^\infty (U)})^ r. \]
At the induction step write $\sigma ^ r=\omega \wedge \sigma ^{r-1} +\mathrm{dd}^{\mathrm{c}}u\wedge \sigma ^{r-1}$, integrate the second term against $\chi $, and move $\mathrm{dd}^{\mathrm{c}}$ to $\chi \sigma ^{r-1}\omega ^{n-r}$. The resulting powers have degrees $r-1,r-2,r-3$, with negative powers absent, and the induction applies on nested slightly larger compact sets. Mixed masses reduce to the single potential average, by positivity of the expansion of its power. Positive mixed currents have total variation controlled by these trace masses.
To prove independence, telescope the difference between two smooth mixed products whose potentials have a common uniform bound. Each term paired with a fixed smooth complementary test is
\[ \int \mathrm{dd}^{\mathrm{c}}(u_ a-v_ a)\wedge T\wedge \eta =\int (u_ a-v_ a)\mathrm{dd}^{\mathrm{c}}(T\wedge \eta ), \]
where $T$ is a product of the remaining admissible factors and $\omega ^{n-m}$. The identity $\mathrm d(\omega +\mathrm{dd}^{\mathrm{c}}u)=\mathrm d\omega $ and its $\mathrm d^{\mathrm c}$ analogue show that derivatives of $T$ introduce only fixed metric derivatives; every term has at most $k-1<m$ admissible factors. The absolute mixed-form bound and the mass estimate just proved bound the variation of $\mathrm{dd}^{\mathrm{c}}(T\wedge \eta )$ uniformly. Consequently the pairing difference is at most $C_\eta \max _ a\| u_ a-v_ a\| _\infty $. The same construction with an arbitrary complementary smooth test also defines the signed products of fewer than $m$ factors without the factor $\omega ^{n-m}$: their variation is bounded by the absolute-form lemma. These products justify the metric-derivative terms above. This proves that all approximations have the same limit and that the products are continuous in the uniform topology. Positivity passes from the smooth mixed products. Using a test supported in an open set on which two limiting potentials coincide proves locality by the same estimate on a smaller neighborhood of its support.
$\square$
Pome
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