We keep the standard small resolution $\pi :X\to Y$ and a smooth positive probability volume form $\mu $ on $X$. Write $\mu _ Y=\pi _*\mu $, and let $\nu _ Y$ be $\omega _ Y^3$ on the regular locus, extended by zero at the nodes. The pullback of the ambient volume form $\omega _ Y^3$ vanishes along the exceptional curves, and its order of vanishing determines the integrability of the density of $\mu _ Y$ with respect to $\nu _ Y$.
Lemma 15.19.14.9225EE111CC9 The measures $\mu _ Y$ and $\nu _ Y$ are finite and give the nodes zero mass. There is a positive density $F$ with $\mu _ Y=F\nu _ Y$. If $\Sigma $ is nonempty, then for $1\leq p<\infty $,
\[ F\in L^ p(Y,\nu _ Y)\quad \Longleftrightarrow \quad p<3. \]
At each node, $F$ is comparable to $r^{-2}$, where $r^2=\sum _{j=1}^4|x_ j|^2$ in the local model $\{ x_1x_4-x_2x_3=0\} $ of the node. There are $c,C,T_0>0$ such that
\[ cT^{-3}\leq \nu _ Y(\{ F>T\} )\leq CT^{-3} \qquad (T\geq T_0). \]
Thus the endpoint is weak $L^3$, with $F\notin L^3$. If $\Sigma $ is empty, $F$ is smooth and bounded.
Proof.
The map is biholomorphic off $E$, and both smooth measures $\mu $ and $\theta ^3$ give $E$ zero mass. Hence $\nu _ Y=\pi _*\theta ^3$ is finite, $\mu _ Y$ has no node atom, and on the regular locus
\[ F\circ \pi =\mu /\theta ^3. \]
In the chart (15.19.9), put $h=1+|z|^2$ and $s=|y_1|^2+|y_2|^2$, so $r^2=hs$. The fiber block of the complex Hessian of $r^2$ is $h$ times the identity, and its Schur complement is $s/h$. Consequently
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15.19.15
\begin{equation} \det \bigl ((r^2)_{j\bar k}\bigr )=hs=r^2, \qquad \frac{(\mathrm{dd}^{\mathrm{c}}r^2)^3}{3!} =\frac{r^2}{\pi ^3}\, \mathrm{d}\lambda _6, \end{equation}
where $\lambda _6$ is Euclidean volume in $(z,y_1,y_2)$. Ambient positivity bounds $\theta $ above and below by positive multiples of $\mathrm{dd}^{\mathrm{c}}r^2$ on a smaller tube. The smooth density of $\mu $ is bounded above and below there. On compact subcharts covering the base $\mathbb P^1$, it follows that
\[ \theta ^3\asymp r^2\, \mathrm{d}\lambda _6, \qquad F\circ \pi \asymp r^{-2}. \]
The other bundle chart gives the same comparison, since $r$ is the norm of the contracted coordinates.
On each bounded base subchart, $r$ is comparable to the Euclidean radius $R$ in the two complex fiber variables. Up to fixed positive constants, the contribution to $\int F^ p\, \mathrm{d}\nu _ Y$ is therefore
\[ \int _0^{R_0}R^{2-2p}R^3\, \mathrm{d}R =\int _0^{R_0}R^{5-2p}\, \mathrm{d}R. \]
Two compact base subcharts give the upper estimate; a fixed open base subchart gives the lower estimate. The integral is finite precisely when $p<3$, and diverges logarithmically at $p=3$. Away from the finitely many nodes the density is smooth. The same comparisons give $\nu _ Y(\{ r<t\} )\asymp t^6$ near a node. The two bounds for $F$ sandwich $\{ F>T\} $ between such tubes of radii comparable to $T^{-1/2}$, proving the endpoint estimate.
$\square$
The descent theorem theorem 9.11.5 applies to the normal space $Y$ and its resolution $\pi $, also when the ambient background is not closed. Every bounded $\theta $-plurisubharmonic function is therefore constant on each exceptional curve and is the pullback of a unique bounded $\omega _ Y$-plurisubharmonic function. In particular each cluster limit has the form $u_0=\psi _0\circ \pi $, with $\sup _ Y\psi _0=0$. Let $\mathrm{MA}^{\mathrm{ext}}_{Y,\omega _ Y}$ be the extended Monge–Ampère measure on $Y$, defined in (9.11.25) as the pushforward of the extended measure of the pullback to a resolution; it does not depend on the resolution. Then
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15.19.16
\begin{equation} \mathrm{MA}^{\mathrm{ext}}_{Y,\omega _ Y}(\psi _0) =c_0\mu _ Y=c_0F\nu _ Y. \end{equation}
The equality holds also at the nodes, since both measures give them zero mass. Thus $\psi _0$ is a bounded solution of an equation on the singular space; nothing is claimed about its continuity at a node.
Theorem 15.19.17.E621E67CCA9A Suppose additionally that $\theta $ is closed. The equation
\[ \mathrm{MA}_\theta (u)=c\mu , \qquad u\in \mathrm{PSH}(X,\theta )\cap L^\infty (X), \qquad \sup _ Xu=0, \]
has a unique pair $(c,u)$ with $c>0$. Its scalar is $c=c_0=\int _ X\theta ^3$. The entire family $u_\varepsilon $ converges to $u$ in $L^1(X)$ and smoothly on compact subsets of $X\setminus E$. The limit is independent of $\eta $, and its descended potential is the unique bounded normalized solution of (15.19.16).
Proof.
Existence follows from corollary 15.19.12. For uniqueness we use an energy identity, which is available because $\theta $ is closed. Fix one cluster limit $u_0$ and put $A=\theta +\mathrm{dd}^{\mathrm{c}}u_0$. This is a positive closed current with bounded local potentials, smooth and strictly positive on $X\setminus E$ by the trace bounds. If $v$ is another bounded solution with scalar $c$, bounded closed-background Stokes gives $c=\int _ X\theta ^3=c_0$. Put $B=\theta +\mathrm{dd}^{\mathrm{c}}v$ and $w=v-u_0$. The bounded integration-by-parts calculus of section 5.27 gives
\[ 0=\int _ Xw(B^3-A^3) =-\sum _{j=0}^2\int _ X\mathrm{d}w\wedge \mathrm d^{\mathrm c}w \wedge B^ j\wedge A^{2-j}. \]
All three energies are nonnegative, so each of them vanishes, in particular the one with $A^2$. On every relatively compact chart of $X\setminus E$, the smooth positive factor $A^2$ controls the weak gradient of $w$, so $\mathrm{d}w=0$ there. The complement of a proper complex analytic subset in the connected manifold $X$ is connected. Thus $w$ is constant almost everywhere off $E$. Since $E$ has ordinary volume zero, canonical almost-everywhere equality of quasi-plurisubharmonic functions, lemma 5.24.3, gives $v=u_0+C$ on all of $X$. Normalization fixes $C=0$.
The compactness and smooth local estimates of theorem 15.19.7 now make every subsequential limit equal to $u_0$, proving convergence of the whole family. The equation and its unique normalized solution do not depend on $\eta $. Descent and the zero exceptional mass identify bounded solutions upstairs and downstairs, proving the final assertion.
$\square$
Pome
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