Proposition 18.38.6B73567245C4F

Proof. In the chart $Z_0=1$ put $z=Z'/Z_0$ and $R^2=a/(1-a)$. For $|z|<R$, the local potential of $\sigma +\mathrm{dd}^{\mathrm{c}}E_ a$ is

Its complex Hessian is semipositive and, away from $z=0$, of rank $q-1$. For $|z|>R$ we have $E_ a=0$, so the form is $\sigma $. The two expressions for $E_ a$ have equal values and equal first derivatives on the sphere $|z|=R$, so they glue to a $C^{1,1}$ function near this sphere, whose distributional Hessian has no part carried by the sphere. Near $b$, $E_ a$ has a logarithmic pole of coefficient $a$. This proves that $E_ a$ is $\sigma $-plurisubharmonic and that $E_ a\sim \psi _ a$.

For the envelope identity, let $w\in \mathrm{PSH}(\mathbb P^ q,\sigma )$ satisfy $w\leq 0$ and $w\preceq _{\mathcal I}\psi _ a$. Since $\psi _ a$ has analytic singularities, lemma 5.38.11 gives $w\leq \psi _ a+C$. Fix a complex line through $0$ in the affine chart. If the restriction of $w+\log (1+|z|^2)$ to this line is identically $-\infty $, there is nothing to prove on the line; otherwise the restriction is subharmonic. On the punctured disk of radius $R$ in the line, subtract $a\log |z|^2$. The difference is subharmonic and, since $w\leq \psi _ a+C$, bounded above near $0$, so it extends subharmonically across $0$. On the boundary circle it is at most $\log (1+R^2)-a\log R^2=-a\log a-(1-a)\log (1-a)$, because $w\leq 0$. By the maximum principle and the formula above, $w\leq E_ a$ on the disk. Every point with $|z|<R$ lies on such a line, and for $|z|\geq R$ we have $w\leq 0=E_ a$. Since $E_ a$ is itself a competitor, it is the envelope. For the continuous obstacle zero, the quasi-everywhere and the pointwise obstacle conditions agree, as recalled in section 5.40.

Off $b$ the local potentials of $\sigma +\mathrm{dd}^{\mathrm{c}}E_ a$ are $C^{1,1}$. By proposition 5.28.5, their Bedford–Taylor top products are given by the determinants of their complex Hessians almost everywhere. By the rank computation, this determinant vanishes where $t<a$, while for $t>a$ the measure is $\sigma ^ q$. The sphere $\{ t=a\} $ has volume zero, and a $C^{1,1}$ local potential puts no mass on it. Finally, the non-pluripolar measure does not charge the point $b$. This proves the measure formula.

For the assertion on $X_ n$, every $\alpha $-plurisubharmonic function has the form $u=w\circ \pi $ with $w\in \mathrm{PSH}(\mathbb P^ q,\sigma )$, by proposition 5.43.5. Since $\varphi $ has analytic singularities, $u\preceq _{\mathcal I}\varphi $ implies $u\preceq \varphi $ by lemma 5.38.11. Restricting this inequality to $\tau =0$ in product coordinates $(z,\tau )$ at $p$ gives $w(z)\leq a\log |z|^2+C$, and hence $w\preceq \psi _ a$ on $\mathbb P^ q$. Conversely, if $w\preceq \psi _ a$, then

near $p$. Away from $p$, the function $\varphi $ is locally bounded and $w$ is bounded above, so by compactness the inequality $w\circ \pi \leq \varphi +C$ holds globally. Thus $w\circ \pi \preceq \varphi $, which implies $w\circ \pi \preceq _{\mathcal I}\varphi $. Since lemma 5.38.11 also applies on the base, the map $u=w\circ \pi \mapsto w$ identifies the competitors on $X_ n$ with the competitors for $E_ a$ considered above. Their largest members therefore correspond under pullback. Finally, theorem 21.14.4 and the measure formula on the base give the last identity. $\square$