PomeLemma 8.12.4.3A8C156268BF Put $k=m-1$ and $\ell =n-m$. For bounded admissible $t_1,\ldots ,t_ k$, let $T=R_ k(t_1,\ldots ,t_ k)\wedge \omega ^\ell $. Then $\mathrm{dd}^{\mathrm{c}}T$ is a signed Radon measure. If bounded admissible $t_ i^ j\downarrow t_ i$ have a common bound, these measures converge weakly, also after multiplication by any continuous function of finitely many uniformly bounded decreasing admissible potentials.
Lemma 8.12.43A8C156268BF
Proof. The subset definition and the torsion formulas in 8.9.7 give, in distributions,
\[ \begin{aligned} \mathrm{dd}^{\mathrm{c}}T={}& \mathrm{dd}^{\mathrm{c}}\omega \wedge \left(\sum _{a=1}^ kR_{k-1}^{\widehat a}\omega ^\ell +\ell R_ k\omega ^{\ell -1}\right)\\ & +2\, \mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega \wedge \sum _{a<b}R_{k-2}^{\widehat a\widehat b}\omega ^\ell \\ & +2\ell \, \mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega \wedge \sum _{a=1}^ kR_{k-1}^{\widehat a}\omega ^{\ell -1}\\ & +\ell (\ell -1)\, \mathrm d\omega \wedge \mathrm d^{\mathrm c}\omega \wedge R_ k\omega ^{\ell -2}. \end{aligned} \]
The sum with two hats is absent when $k=1$, and the last term is absent when $\ell =1$. Every factor $R_ r$ has degree $r\leq k<m$, so its coefficients are signed measures. Expanding each $R_ r$ over subsets expresses the displayed measure as fixed smooth forms wedged with raw currents $D_ p$ of degrees $p\leq m-1$. 8.11.10 therefore bounds its variation locally by $C c_{m-1}^*$, uniformly under common bounds on the potentials. Joint convergence follows from 8.11.12 applied to this expansion. All the canonical weights are quasicontinuous for $c_{m-1}^*$. 8.11.11, with $s=m-1$, thus proves the weighted assertion coefficientwise on a finite coordinate cover. A partition of unity gives it on $X$. The term involving $R_ k\mathrm{dd}^{\mathrm{c}}(\omega ^\ell )$ retains degree $m-1$; the argument requires no coefficient estimate in degree $m$. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.