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We now prescribe a singularity type with logarithmic poles. If the poles become a divisor on a modification, the singular part can be removed there, and what remains is a smooth residual form. When this residual form is strictly positive, the bounded Hermitian theory on the modification provides the remaining bounded potential, and pushing forward solves the equation in the prescribed type.
In this section $n\geq 2$, and the background is a Hermitian form $h$ on $X$. Let $\psi \leq 0$ be a quasi-plurisubharmonic function that is smooth outside a nonempty analytic set $A$, its pole set. Assume that there is a modification $\pi :Y\to X$, biholomorphic over $X\setminus A$, such that $\pi ^{-1}(A)$ is the support $|D|$ of a simple normal crossing divisor $D=\sum _ jD_ j$ and
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\begin{equation} \psi \circ \pi =\sum _ j c_ j\log |s_ j|^2+b, \qquad \beta =\pi ^*h+\mathrm{dd}^{\mathrm{c}}b-\sum _ jc_ j\Theta _ j>0, \end{equation}
where $c_ j>0$, $b$ is smooth on $Y$, $s_ j$ is a holomorphic section of $\mathcal O(D_ j)$ defining $D_ j$, and $\Theta _ j$ is the curvature form of a smooth Hermitian metric on $\mathcal O(D_ j)$, so that $\mathrm{dd}^{\mathrm{c}}\log |s_ j|^2=[D_ j]-\Theta _ j$. Thus $\beta $ is the smooth part of the current $\pi ^*h+\mathrm{dd}^{\mathrm{c}}(\psi \circ \pi )$, which we call the residual form.
Theorem 15.22.2.3B5397268E44 For every nonzero nonnegative $f\in L^ p(X,h^ n)$, $p>1$, there are $\kappa >0$ and a normalized $u\in \mathrm{PSH}(X,h)$ such that
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\begin{equation} \mathrm{MA}^{\mathrm{ext}}_ h(u)=\kappa f h^ n, \qquad u=\psi +O(1). \end{equation}
The lifted remainder $v=(u-\psi )\circ \pi $ extends Hölder continuously over $Y$, so $u$ is locally Hölder on $X\setminus A$. Its current is
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\begin{equation} \pi ^*h+\mathrm{dd}^{\mathrm{c}}(u\circ \pi ) =\sum _ jc_ j[D_ j]+\beta +\mathrm{dd}^{\mathrm{c}}v. \end{equation}
The positive residual current $\beta +\mathrm{dd}^{\mathrm{c}}v$ has zero Lelong numbers. If $h$ is Kähler, then $\kappa =\int _ Y\beta ^ n/\int _ X f h^ n$, and the normalized solution is unique among all normalized solutions $u'$ of (15.22.3) with $u'=\psi +O(1)$, continuous or not. If the density $g=(f\circ \pi )\, \pi ^*h^ n/\beta ^ n$ is smooth and strictly positive on $Y$, the residual current $\beta +\mathrm{dd}^{\mathrm{c}}v$ may be chosen smooth and strictly positive.
Proof.
Define $J=\pi ^*h^ n/\beta ^ n$ and $g=(f\circ \pi )J$ off the exceptional set, assigning any value to $g$ on that volume-zero set. The smooth nonnegative $J$ is bounded. Change of variables on the biholomorphic locus gives
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\begin{equation} \int _ Yg\beta ^ n=\int _ Xfh^ n, \qquad \int _ Yg^ p\beta ^ n\leq \| J\| _\infty ^{p-1}\int _ Xf^ ph^ n. \end{equation}
Thus $g$ is a nonzero $L^ p$ density. Theorem 14.19.2, Theorem 15.5.1 give a bounded Hölder $\beta $-plurisubharmonic $v$ and $\kappa >0$ with $(\beta +\mathrm{dd}^{\mathrm{c}}v)^ n=\kappa g\beta ^ n$. If $h$ is Kähler, then $\beta $ is Kähler and the mass identity fixes the stated scalar. For smooth positive $g$, theorem 14.2.1 gives a smooth solution instead.
Set $u=\psi +v\circ \pi ^{-1}$ on $X\setminus A$ and $u=-\infty $ on $A$. Push forward the positive current $\sum c_ j[D_ j]+\beta +\mathrm{dd}^{\mathrm{c}}v$. Pushforward commutes with $\mathrm{dd}^{\mathrm{c}}$, $\pi _*\pi ^*h=h$, and the pushed locally integrable potential is $u$ almost everywhere. Its quasi-plurisubharmonic representative equals $\psi +v\circ \pi ^{-1}$ off $A$ and tends to $-\infty $ at $A$ because $v$ is bounded. Hence $h+\mathrm{dd}^{\mathrm{c}}u\geq 0$ and $u=\psi +O(1)$ in the two-sided ordinary sense. The pulled-back identity is (15.22.4); bounded residuals have zero Lelong number. The equation holds off $A$ by (15.22.5), and theorem 13.14.3 extends it as (15.22.3). Subtracting $\sup _ Xu$ changes neither the equation nor the ordinary type.
For uniqueness when $h$ is Kähler, let $u'$ be a normalized solution with $u'=\psi +O(1)$. On $Y\setminus D$, $w=(u'-\psi )\circ \pi $ is bounded and $\beta $-plurisubharmonic. Since $w$ is bounded, lemma 5.24.7 extends it across $D$ to a bounded $\beta $-plurisubharmonic function on $Y$. On each relatively compact subset of $Y\setminus D$, sufficiently deep truncations of $u'$ equal $u'$. Thus locality in theorem 13.14.3 gives the bounded equation $\kappa g\beta ^ n$ there; both sides give $D$ zero mass. The global equation follows, its mass fixes $\kappa $, and closed full-mass uniqueness gives $w=v+$constant as recalled in section 25.21. Normalization fixes that constant.
$\square$
The Hölder constants depend on the fixed geometry and divisor metrics, the $c_ j$, a lower bound for the positivity of $\beta $, $p$, and the $L^ p$ norm of the density $\kappa g$ itself. In particular, the argument gives no estimate that is uniform as $\beta $ degenerates or as $\kappa $ becomes unbounded. In simple normal crossing coordinates a density comparable to $\prod _ j|z_ j|^{-2a_ j}$ belongs locally to $L^ p$ exactly when
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\begin{equation} pa_ j<1\quad \hbox{for every }j, \end{equation}
by the factors $\int _0^\epsilon r_ j^{1-2pa_ j}\, \mathrm{d} r_ j$. A smooth positive density $f$ on $X$ need not give a strictly positive $g$ on $Y$, since the Jacobian factor $J=\pi ^*h^ n/\beta ^ n$ of the proof can vanish on the exceptional set.
Example 15.22.7.0362BBF11A0B Take $X=\mathbb P^ n$, $h=\lambda \omega _{\mathrm{FS}}$, $\int _ X\omega _{\mathrm{FS}}^ n=1$, and $0<c<\lambda $. Put
\[ \psi _ c=c\log \frac{\sum _{j=1}^ n|Z_ j|^2}{\sum _{j=0}^ n|Z_ j|^2}. \]
Its pole is $P=[1:0:\cdots :0]$. On the point blowup $\pi :Y\to X$, let $q:Y\to \mathbb P^{n-1}$ be the direction map. In affine blowup coordinates $x=(z,zw)$, Poincaré–Lelong gives
\[ \pi ^*h+\mathrm{dd}^{\mathrm{c}}(\psi _ c\circ \pi ) =c[E]+(\lambda -c)\pi ^*\omega _{\mathrm{FS}}+cq^*\omega _{\mathrm{FS}}. \]
The residual form $\beta =(\lambda -c)\pi ^*\omega _{\mathrm{FS}}+cq^*\omega _{\mathrm{FS}}$ is positive: $\pi ^*\omega _{\mathrm{FS}}$ is positive in the directions normal to $E$, and $q^*\omega _{\mathrm{FS}}$ in the directions tangent to $E$. Write $H=\pi ^*[\omega _{\mathrm{FS}}]$. Since $H|_ E=0$ and $E^ n=(-1)^{n-1}$, its mass is
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\begin{equation} \int _ Y\bigl ((\lambda -c)\pi ^*\omega _{\mathrm{FS}}+cq^*\omega _{\mathrm{FS}}\bigr )^ n=\lambda ^ n-c^ n. \end{equation}
Hence, for every nonzero nonnegative $f\in L^ p$ with $\int _ Xfh^ n=\lambda ^ n-c^ n$, theorem 15.22.2 gives $\kappa =1$ and a solution of $\mathrm{MA}^{\mathrm{ext}}_ h(u)=fh^ n$ with the nonminimal type $u=\psi _ c+O(1)$.
For an explicit solution, choose a smooth function $v$ on $Y$ with $v|_ E$ nonconstant, small enough that $S=\beta +\mathrm{dd}^{\mathrm{c}}v$ is still positive, and put $u=\psi _ c+v\circ \pi ^{-1}$ off $P$. The density $f$ of its measure, given by $f\circ \pi =S^ n/\pi ^*h^ n$ off $E$, lies in every $L^ p$ with $1<p<n/(n-1)$: $\pi ^*h^ n$ vanishes to order $|z|^{2(n-1)}$ along $E$, so the local integral of $f^ p$ is finite precisely when $(n-1)(p-1)<1$. Different directions of approach to $P$ give different limits of $u-\psi _ c$. The divisor $c[E]$ of the current upstairs leaves no divisor downstairs, since $E$ is contracted to a point. On $X\setminus \{ P\} $, where $u$ is smooth and strictly $h$-plurisubharmonic, the Hermitian metric $h+\mathrm{dd}^{\mathrm{c}}u$ is isometric to $(Y\setminus E,S)$. Its metric completion is $Y$ with the metric $S$, so the whole divisor $E$ is restored: removing the smooth real codimension-two submanifold $E$ does not change the infimum of path lengths in $Y$. Finally, if $h$ is replaced by a sufficiently small nonclosed perturbation, both $h$ and the residual form remain positive, so the existence part of theorem 15.22.2 still applies.
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