Lemma 10.17.321B7909B2B52

Proof. Off the support $|\Gamma |$, we have $\mathrm{dd}^{\mathrm{c}}\ell =-\sum _ jc_ j\gamma _ j$, so the current $\alpha _ Z+\mathrm{dd}^{\mathrm{c}}v_\pi $ equals $\eta _ v+\mathrm{dd}^{\mathrm{c}}w$ there and is positive. After the addition of a smooth local potential for $\eta _ v$, the bounded function $w$ is plurisubharmonic off $|\Gamma |$. By lemma 5.24.7, it extends across the divisor as a bounded plurisubharmonic function that agrees almost everywhere with the original one; we use this extension to represent $w$. Thus $R_ v\geq 0$ on $Z$, and the Poincaré–Lelong formula gives

Since $w$ is bounded, $b_ v$ is nef by lemma 20.4.17. The non-pluripolar measure on the left of (10.17.4) agrees with $R_ v^ k$ off the divisor, and neither measure charges the divisor. This gives the first equality.

Fix a Kähler form $\kappa $ on $Z$. For all sufficiently large real $t$, the smooth closed form $\eta _ v+t\kappa $ is Kähler, and $w$ is a bounded potential for it. Therefore

by theorem 11.1.7, or by Stokes' theorem when $k=1$. Both sides are polynomials in $t$, so their constant coefficients agree; this gives $\int _ ZR_ v^ k=\int _ Zb_ v^ k$. The last equality in (10.17.4) is (10.16.6). Applying the same polynomial argument to positive linear combinations of the $R_{v_ i}$ and comparing coefficients gives (10.17.5). $\square$