Lemma 15.19.10C6428C798D5F

Proof. Smoothness and positivity off $E$ follow from the pullback and the biholomorphism. In one standard model put

The function $r^2$ is the sum of the squared absolute values of the four contracted coordinates. It is therefore globally defined on the bundle; $b$ is the pullback of the Fubini–Study form from its base. In the chart (15.19.9),

This is an inequality of currents across the zero section as well as a smooth inequality off it.

Ambient positivity gives $\theta \geq c\mathrm{dd}^{\mathrm{c}}r^2$ on a sufficiently small tube, for some $c>0$. The smooth form $\mathrm{dd}^{\mathrm{c}}r^2+b$ is positive definite there. Indeed the contraction is an immersion off the zero section, and on that section $\mathrm{dd}^{\mathrm{c}}r^2$ is positive in both fiber directions while $b$ is positive along the base. Compactness of the zero section and a smaller closed tube give a uniform lower bound by a positive multiple of $\eta $. Thus, for every fixed $a>0$,

near the zero section, with $\delta _ a>0$.

Choose disjoint model tubes with nonempty complement and numbers $0<t_1<t_2<1$ in each. Let $0\leq \chi \leq 1$ be a smooth cutoff of $r^2$, equal to one for $r^2\leq t_1$ and zero for $r^2\geq t_2$. Extend $\chi (r^2)\log r^2$ by zero outside its tube. On the compact transition annuli, the pullback $\theta $ is uniformly positive and the second derivatives of this cutoff potential are bounded. A sufficiently small positive rational $a$ therefore makes

satisfy $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \delta \eta $ there and near every $E_ j$. On the remaining compact set, $\rho =0$ and $\theta $ is positive, so decreasing $\delta $ proves the global inequality. Near $E_ j$ the function is $a\log (|y_1|^2+|y_2|^2)+a\log (1+|z|^2)$, hence has analytic singularities with rational coefficient and pole set exactly $E_ j$. It is smooth elsewhere and nonpositive; its zero extension gives supremum zero. If $E$ is empty, take $\rho =0$ and use the positivity of $\theta $ on compact $X$. $\square$