Theorem 15.23.10.37D35D78F0C6 Let $\pi :Y_ m\to Y_0$ be a finite nonempty sequence of blowups of connected smooth centers of codimensions $r_ j\geq 2$ over a compact Hermitian $(Y_0,h_0)$. Let $E_ j$ be the exceptional divisor at stage $j$, let $\pi _{>j}$ denote the subsequent map, choose $a_ j>0$, and set
\[ \Delta =\sum _ j a_ j\pi _{>j}^*E_ j, \qquad \mathrm{dd}^{\mathrm{c}}L_\Delta =[\Delta ]-\Theta _\Delta , \qquad \theta =\pi ^*h_0+\Theta _\Delta , \]
where $L_\Delta =\sum _ ja_ j\log |s_ j|^2$, with $s_ j$ the canonical section of $\pi _{>j}^*\mathcal O(E_ j)$ for a smooth Hermitian metric, and $\Theta _\Delta $ is the corresponding combination of curvature forms. Every $\theta $-positive current is $\pi ^*(h_0+\mathrm{dd}^{\mathrm{c}}q)+[\Delta ]$ and every potential is $L_\Delta +q\circ \pi +C$, with $q\in \mathrm{PSH}(Y_0,h_0)$. Minimal ordinary type is exactly $L_\Delta +O(1)$, equivalent to bounded $q$. For potentials of this type, every prime exceptional coefficient is the one prescribed by the effective total transform $\Delta $. There is a strict barrier with divisor $\Delta +B$ for an effective exceptional $B$ positive on every exceptional component. For nonzero nonnegative upstairs $L^ p$ data, $p>1$, a minimal solution has Hölder $q$, with pushed density in every $L^ s$ satisfying
749BE96956FE
15.23.11
\begin{equation} 1<s<\frac{Rp}{(R-1)p+1},\qquad R=\prod _ jr_ j. \end{equation}
For Kähler $h_0$ the mass, scalar, and closed full-mass uniqueness are those of the target; the power conditions on $h_0$ fix the scalar for the constructed nonclosed solution. For the pullback measure $\pi ^*(h_0^ n)$, the metric $\pi ^*h_0$ on the complement of the exceptional set has metric completion $Y_0$, each fiber collapsing to a point, and the zero divisor of the density is that of the squared modulus of the holomorphic Jacobian, independently of the $a_ j$.
Proof.
Apply proposition 15.23.2 successively, allowing an intermediate smooth background to be indefinite. The difference between each current and the pullback of its pushforward is closed of order zero, and exceptional-fiber degree fixes its coefficient. Pullback of earlier divisor currents produces the total transforms, proving the representation without an SNC requirement. The initial $q=0$ is admissible; upper bounds on compact targets prove minimality and the bounded-$q$ criterion. At SNC charts this is the usual sum of logarithms.
At each stage choose an auxiliary divisor metric as in proposition 15.23.2, with curvature $\Theta '_{E_ j}$, so that $h_ j=\pi _ j^*h_{j-1}-\delta _ j\Theta '_{E_ j}>0$ for a sufficiently small $\delta _ j>0$. Expanding yields $h_ m=\pi ^*h_0-\Theta _ B$ for $B=\sum _ j\delta _ j\pi _{>j}^*E_ j$, effective with a positive coefficient on every exceptional component. If $\mathrm{dd}^{\mathrm{c}}L_ B=[B]-\Theta _ B$ uses these auxiliary metrics, then the fixed background satisfies $\theta +\mathrm{dd}^{\mathrm{c}}(L_\Delta +L_ B)=h_ m+[\Delta +B]$, which is the strict barrier current. The Jacobian calculation and Hölder estimate apply stagewise. If $s=rp/((r-1)p+1)$, then $1-1/s=(1-1/p)/r$; iteration over the stages gives (15.23.11), with strict inequality. The target Hölder equation, mass and uniqueness are obtained as in theorem 15.23.5. For the pullback measure take $q=0$ and use the target path comparison away from the exceptional set. The Jacobian fixes the density, while $[\Delta ]$ fixes the divisor current; these are independent pieces of the equation.
$\square$
Pome
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