Pome91E1AFBF43BECorollary 18.26.4.91E1AFBF43BE Fix $x$ and the models $p_ a$ of proposition 18.26.2. Let $\pi :Y\to X$ be the blowup at $x$, with exceptional divisor $E$, and let $\eta $ be a Hermitian form on $Y$. Then
For $\mu _ a=\operatorname {Pol}_\pi (p_ a)\wedge \eta ^{n-1}$ and any two admissible coefficients $a,b$,
Proof. On relatively compact blowup charts, with $z=(s,sw)$, one has $p_ a\circ \pi =a\log |s|^2+O(1)$ along $E=\{ s=0\} $. Thus the pole set is $E$ and the generic Lelong number along it is $a$. Add a local smooth function $\rho $ with $\mathrm{dd}^{\mathrm{c}}\rho \geq \pi ^*\omega $. By the argument of lemma 5.38.1, the restriction of the closed positive current $\mathrm{dd}^{\mathrm{c}}(p_ a\circ \pi +\rho )$ to $E$ is $a[E]$, and subtracting the smooth correction does not change the coefficient mass on $E$. Consequently $\mu _ a=a[E]\wedge \eta ^{n-1}$. The positive measure $[E]\wedge \eta ^{n-1}$ has mass $\int _ E\eta ^{n-1}>0$, which proves the formula for the total variation. $\square$
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