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.
$\square$
Pome
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