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
4669708E978C
15.22.3
\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
91C08F2AC112
15.22.4
\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
BBA63F88F7F7
15.22.5
\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.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.