Proposition 16.29.187F133849FE58

Proof. The potential $u_\eta =\psi _ F+f_\eta $ has analytic singularities, so proposition 18.4.8 gives $q_\eta \sim u_\eta \sim \psi _ F$. Removing the divisorial component $[F]$ gives a positive current in the background $\theta +\eta $, whose global potential $w_\eta =q_\eta -\psi _ F$ is bounded by this ordinary equivalence.

Off $F$, plurifine locality identifies the Monge–Ampère measure of $q_\eta $ with $(\theta +\eta +\mathrm{dd}^{\mathrm{c}}w_\eta )^2$. This bounded product gives no mass to $F$ by proposition 5.28.2. For closed $\eta =\epsilon \kappa $, Stokes' theorem gives its total mass as $\int _ X(\theta +\epsilon \kappa )^2$.

The normal bundle of $D$ is trivial. Indeed, the transformation $w\mapsto w+1$ becomes $\xi \mapsto \xi /(1+\xi )$, whose derivative at $\xi =0$ is one, and the other transformation fixes $w$. Hence $\int _ X\theta ^2=D^2=0$, and expanding the square gives the formula. The limiting potential has curvature current $[F]+[D]$, which vanishes away from the pluripolar set $F\cup D$; its non-pluripolar measure is therefore zero. $\square$