Lemma 15.16.1154DE11154EE

Proof. Let $\pi :Y\to X$ be a modification resolving the singularities of $\psi $. The pulled-back current has the form

where $D$ is an effective real divisor, $[D]$ is its current of integration, and $\beta $ is a smooth real $(1,1)$-form on $Y$. Off $D$ we have $\beta \geq a\pi ^*\omega $, and by continuity this inequality holds everywhere. In particular, $\beta $ is semipositive.

We check that $\beta $ is big. For a blowup with smooth center and exceptional divisor $E$, choose a metric on $\mathcal O(-E)$ whose curvature is positive in the directions tangent to the projective fibers. Along $E$, the pullback of a Hermitian metric is positive in the complementary directions. Consequently, its sum with a sufficiently small positive multiple of this curvature form is a Hermitian metric: positivity holds first along $E$, then on a neighborhood of $E$, and finally on the compact complement, where the pullback is already positive. Iterating over the blowups, we obtain an effective real exceptional divisor $F$, with logarithmic potential $\zeta $, and a smooth Hermitian form $\kappa $ such that

At each step of the iteration, we pull back the smooth positive form obtained so far, apply the construction for one blowup, and add the pullback of the previous divisor to the new positive multiple of $E$. Hence $\beta +a\mathrm{dd}^{\mathrm{c}}\zeta \geq a\kappa $, so $\beta $ is big. This argument only involves Hermitian metrics and never requires a closed positive form on $Y$.

Fix a smooth positive volume form $\mathrm{d}V_ Y$ on $Y$, and write $\pi ^*\mathrm{d}V=J\, \mathrm{d}V_ Y$. The function $J$ is smooth, nonnegative and bounded, and it is positive off the exceptional locus. Define

This function is nonzero and belongs to $L^ p(\mathrm{d}V_ Y)$. Indeed, $\psi $ is bounded above, and the change of variables formula gives

the analytic exceptional sets do not affect these integrals. Theorem 15.3.3, applied on $Y$ in the semipositive big background $\beta $, gives a bounded $s\in \mathrm{PSH}(Y,\beta )$ with

The function $\pi ^*\psi +s$ is $\pi ^*\theta $-plurisubharmonic, because the corresponding current is $\beta _ s+[D]\geq 0$, where $\beta _ s=\beta +\mathrm{dd}^{\mathrm{c}}s$. By theorem 9.11.5 it descends to some $u\in \mathrm{PSH}(X,\theta )$, and $u\sim \psi $ because $s$ is bounded. Off the exceptional set and the pole set of $\psi $, the change of variables formula gives the required equation. Neither the extended measure on the left nor the measure $\mathrm{e}^{\lambda u}f\, \mathrm{d}V$ on the right charges these analytic sets, so the equation holds on all of $X$. $\square$