PomeLemma 12.18.6.5771D42397D7 For a fixed positive current $T$ and fixed $\mathrm dV$, the measure $\Gamma _{0,h}(T)$ is independent of the positive auxiliary metric $h$. This statement includes singular and degenerate currents.
Lemma 12.18.65771D42397D7
Proof. In a holomorphic frame put $\lambda =\bigwedge _{j=1}^ n(i\, dz_ j\wedge d\bar z_ j)$, $h=i\sum H_{ij}\, dz_ i\wedge d\bar z_ j$ and $\mathrm dV=v\lambda $. Positive currents have measure coefficients. Choose a positive Radon measure $\nu $ dominating them. For a smooth real form $\xi =i\sum \Xi _{ij}\, dz_ i\wedge d\bar z_ j$, let $Q$ be the positive semidefinite Hermitian coefficient matrix characterized by $\xi \wedge T=\operatorname {tr}(\Xi Q)\nu $. Then
\[ \sigma _ h=\operatorname {tr}(HQ)\nu ,\qquad P_ h=\frac{nH^{1/2}QH^{1/2}}{\operatorname {tr}(HQ)}. \]
The formula holds on $Q\ne 0$; zero is assigned when $Q=0$. Since $h^ n=n!\det (H)\lambda $, determinant algebra gives
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12.18.7
\begin{equation} \Gamma _{0,h}(T) =n(n!)^{-1/n}v^{1/n}(\det Q)^{1/n}\nu . \end{equation}
This expression has no $H$. Replacing $\nu $ by $L\nu $, where $L>0$ almost everywhere, replaces $Q$ by $Q/L$, leaving the measure unchanged; a common dominating measure compares any two choices. The original geometric definition ensures compatibility under frame changes. Matrix AM–GM bounds it locally by a constant times a reference trace measure, proving local finiteness. Compactness gives a finite Radon measure. $\square$
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