Proposition 15.23.2E42F32DA29D8

Proof. Put $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ and $q=\pi _*(u-cL)$ as a scalar distribution. The latter is locally integrable: off $Z$ it is the inverse-coordinate function, and a smooth target volume pulls back with bounded Jacobian, so its absolute integral is controlled by that of $u-cL$ on $X$. Since $\pi _*[E]=0$,

A smooth local strictly plurisubharmonic shift gives a quasi-plurisubharmonic representative of $q$. Its pullback $q\circ \pi $ is not identically $-\infty $ and satisfies $\pi ^*(h+\mathrm{dd}^{\mathrm{c}}q)\geq 0$, by lemma 5.43.2: since $\pi $ is dominant, the composition of a local plurisubharmonic function with $\pi $ is again plurisubharmonic and not identically $-\infty $.

Off $E$, $T=\pi ^*S$. The difference is an order-zero current supported on $E$, and it is closed: both derivatives equal $\pi ^*\mathrm{d} h$. To identify it, write $R=T-\pi ^*S$. In a chart where $E=\{ z=0\} $, order zero gives $zR=\bar zR=0$. Closedness then gives $\mathrm{d}z\wedge R=\mathrm{d}\bar z\wedge R=0$. Thus only its normal $(1,1)$ coefficient remains, a measure on $E$; the tangential equations in $\mathrm{d}R=0$ make that measure a constant multiple of integration on $E$. Connectedness of $E$ identifies one constant $a$ globally, so $R=a[E]$. Its cohomology class is $c[E]$. On a line in an exceptional fiber, $\mathcal O(E)$ has degree $-1$, so $[E]$ is nonzero and $a=c$. The difference between the two scalar potentials in (15.23.3) has $\mathrm{dd}^{\mathrm{c}}$ zero. Local elliptic regularity makes it pluriharmonic; compactness and connectedness make it constant.

Since $h>0$, $q=0$ is admissible, and every $h$-plurisubharmonic $q$ is bounded above on compact $Y$. Hence $cL$ is minimal and two-sided bounded comparison with it is exactly boundedness of $q$. For the barrier choose another divisor metric, with logarithmic norm $L'=L+\sigma $ and curvature $\Theta '_ E=\Theta _ E-\mathrm{dd}^{\mathrm{c}}\sigma $ for a smooth real function $\sigma $, so that $-\Theta '_ E$ is positive on the vertical tangent bundle of $E\to Z$, using the tautological $\mathcal O(-1)$. In adapted coordinates

$\pi ^*h$ controls the nonvertical directions and $|z|^2|\mathrm{d} w|^2$. For small $\delta >0$, the vertical positivity of $-\delta \Theta '_ E$ dominates its mixed errors by completing the square; away from $E$, compactness allows the same $\delta $. Thus $\pi ^*h-\delta \Theta '_ E>0$. For the fixed background $\theta $ in (15.23.1), take

Then $\theta +\mathrm{dd}^{\mathrm{c}}w=(c+\delta )[E]+\pi ^*h-\delta \Theta '_ E$ dominates a positive Hermitian form. The smooth correction $\delta \sigma $ preserves the stated logarithmic coefficient. $\square$