18.38 A point pole on the Hopf manifoldCFB70C5F514C

Proof. Since $n=q+1$, proposition 18.37.3 identifies the selected space with the space of homogeneous polynomials of degree $m$ on $\mathbb C^{q+1}$ that vanish to order at least $r_ m$ at $b$. For $\ell =(\ell _0,\ldots ,\ell _ q)\in \mathbb Z_{\geq 0}^{q+1}$ write $|\ell |=\sum _ i\ell _ i$ and $Z^\ell =\prod _ iZ_ i^{\ell _ i}$. In the affine chart $Z_0=1$, the vanishing order of $Z^\ell $ at $b$ is $\ell _1+\cdots +\ell _ q$, so the selected space is spanned by the monomials $Z^\ell $ with $|\ell |=m$ and $\ell _1+\cdots +\ell _ q\geq r_ m$.

Distinct monomials are orthogonal, by invariance under the coordinate circle actions. Their squared norms with respect to $\sigma ^ q$ are

To verify this, integrate $|z^\ell |^2\mathrm{e}^{-|z|^2}$ over $\mathbb C^{q+1}$ in two ways. As a product of one-variable integrals it equals $\pi ^{q+1}\prod _ i\ell _ i!$, while in polar coordinates it is the average of $|z^\ell |^2$ over the unit sphere times a radial integral that contributes the factor $(m+q)!$. Dividing by the same computation for $\ell =0$ gives the right-hand side of (18.38.3) as the normalized sphere average, and the pushforward of the normalized sphere measure to $\mathbb P^ q$ is $\sigma ^ q$, since both are the probability measures invariant under the unitary group. This proves (18.38.3).

The reciprocal of the squared norm is $D_ mm!/(\ell _0!\cdots \ell _ q!)$. Summing first over the monomials with $\ell _1+\cdots +\ell _ q=j$, the multinomial formula gives the contribution

to the kernel on $\mathbb P^ q$. Since $\nu _ X$ is the lift of $\sigma ^ q$, the norms and kernels lift exactly to $X_ n$, which proves the kernel formula. There are $\binom {j+q-1}{q-1}$ monomials of affine degree $j$, and summing over the excluded degrees $0\leq j<r_ m$ gives $\binom {r_ m+q-1}{q}$. Since $0\leq r_ m\leq m$ and $r_ m/m\to a$, this proves the dimension formula, the nonvanishing and the limit. $\square$

Proof. Write $P_ m(t)$ for the binomial sum in proposition 18.38.2. For fixed $0<t<1$, $P_ m(t)$ is the probability that a binomial random variable with parameters $(m,t)$ is at least $r_ m$. Its variance is $mt(1-t)$, so Chebyshev's inequality and $r_ m/m\to a$ give

The pushforward of $\sigma ^ q$ under $t$ is $q t^{q-1}\, \mathrm{d}t$. To see this, note that in the affine chart the Fubini–Study measure has radial density $2q r^{2q-1}(1+r^2)^{-q-1}\, \mathrm{d}r$, and $t=r^2/(1+r^2)$. In particular, the level set $\{ t=a\} $ has measure zero. Since $0\leq P_ m\leq 1$ and $q!D_ m/m^ q\to 1$, dominated convergence gives convergence of the densities in $L^1(\sigma ^ q)$, hence in $L^1(\nu _ X)$ after lifting; this is the asserted convergence in total variation. The limit has mass $\int _ a^1 qt^{q-1}\, \mathrm{d}t=1-a^ q$.

To identify the limit of the potentials, put

with $0\log 0=0$. Then

Indeed, $\binom mj\mathrm{e}^{-mH(j/m)}$ is the $j$th term of the binomial distribution with parameters $(m,j/m)$. It is at most one, and it is at least $1/(m+1)$ because, as the ratio of neighboring terms shows, it is a largest of the $m+1$ terms, which sum to one. For $j=0$ and $j=m$ it equals one.

The logarithm of a sum of at most $m+1$ nonnegative terms differs from the largest of their logarithms by at most $\log (m+1)$. Hence $m^{-1}\log P_ m(t)$ differs by at most $\log (m+1)/m$ from

On compact subintervals of $(0,1)$, uniform continuity and $r_ m/m\to a$ show that these maxima converge uniformly to the maximum of the same expression over $a\leq s\leq 1$. Maximizing in $s$, we find that this maximum is zero for $t\geq a$, attained at $s=t$, and equals the first expression in (18.38.1) for $t<a$, attained at $s=a$. Thus the limit is $e_ a(t)$, also at $t=a$.

The convergence is also uniform near $t=1$. Choose $c$ with $a<c<1$. The same variance estimate gives $P_ m(t)\geq 1-C/m$ uniformly for $c\leq t\leq 1$ and large $m$, so $m^{-1}\log P_ m\to 0$ uniformly there. Together with the argument on compact subintervals and $m^{-1}\log D_ m\to 0$, this proves locally uniform convergence off $\{ t=0\} $. At $t=0$ we have $r_ m>0$ for large $m$, so $P_ m(0)=0$ and the pointwise limit is $-\infty $.

Finally, $t^ m\leq P_ m(t)\leq 1$, because the term $j=m$ occurs in every binomial sum. Since $m^{-1}\log D_ m$ is nonnegative and bounded, it follows that $\log (t\circ \pi )\leq U_ m\leq C$ off the fiber over $b$. In local submersion coordinates near this fiber, $\log (t\circ \pi )=\log |z|^2+O(1)$, which is integrable against a smooth volume. The pointwise convergence and these bounds give convergence in $L^1$, by dominated convergence. $\square$

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. $\square$