Example 18.39.17.17FFBB442DEB Let $q\geq 2$, $B=\mathbb P^ q$, $A=\mathcal O_ B(1)$, $p=[1:0:\cdots :0]$ and $X=B\times \mathbb P^1$. Use the normalized Fubini–Study forms $\omega _ B,\omega _ F$ and the corresponding metrics, and for a point $o\in \mathbb P^1$ put
\[ R=\frac{\sum _{j=1}^ q|Z_ j|^2}{\sum _{j=0}^ q|Z_ j|^2},\quad T=\| s_ o\| ^2,\quad \varphi =\tfrac 14\log (R^2+T^3),\quad \psi =\tfrac 12\log R. \]
Then $\varphi \in \mathrm{PSH}(X,\omega _ B+\omega _ F)$, and $\psi \in \mathrm{PSH}(B,\omega _ B)$ has mass $1-2^{-q}$. Let $\mathfrak m_ p$ be the ideal sheaf of $p$. The selected and reference spaces are exactly
\[ \begin{gathered} V_ m=H^0\bigl (B,A^ m\otimes \mathfrak m_ p^{k_ m}\bigr ),\qquad W_ m=H^0\bigl (B,A^ m\otimes \mathfrak m_ p^{\ell _ m}\bigr ),\\ k_ m=\max \{ \lfloor m/2-2/3\rfloor -q+1,0\} ,\qquad \ell _ m=\max \{ \lfloor m/2\rfloor -q+1,0\} . \end{gathered} \]
Their defect vanishes for $m<2q$, and for $m\geq 2q$ it is
\[ \dim V_ m-\dim W_ m=\binom {\lfloor m/2\rfloor -1}{q-1} \sim \frac{2^{1-q}}{(q-1)!}m^{q-1}. \]
Proof.
Over a common denominator, $R^2+T^3$ is a sum of squared moduli of holomorphic coefficients of sections of $\mathcal O_ X(2,3)$, divided by the corresponding metric factor. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -\tfrac 12\omega _ B-\tfrac 34\omega _ F$, also across the common zero set by the extension theorem.
Let $\rho :Y\to B$ be the blowup of $p$, with exceptional divisor $E$, and let $\tau :Y\to \mathbb P^{q-1}$ be the extension to $Y$ of the projection $[Z_0:\cdots :Z_ q]\mapsto [Z_1:\cdots :Z_ q]$. Since $(\rho ,\tau )$ is an embedding, the forms
\[ \alpha =\rho ^*\omega _ B,\quad \gamma =\tau ^*\omega _{\mathbb P^{q-1}},\qquad \beta =\tfrac 12(\alpha +\gamma ) \]
satisfy $\beta >0$. The divisor bundle is $\mathcal O_ Y(E)=\rho ^*\mathcal O_ B(1)\otimes \tau ^*\mathcal O_{\mathbb P^{q-1}}(-1)$; its quotient metric has curvature $\alpha -\gamma $ and satisfies $R\circ \rho =\| s_ E\| ^2$. Thus the theorem applies with the single divisor $E$, $u=0$, $c=1/2$, $a=2/3$ and residual $\beta $. Along $E\times \{ o\} $ the pulled-back selecting potential has the exact local model $\tfrac 14\log (|z|^4+|w|^6)+O(1)$, and it is locally bounded elsewhere. If $T_ m(s)$ is divisible by $z^{K_ m}$, where $K_ m=\max \{ \lfloor m/2-2/3\rfloor ,0\} $, then lemma 18.39.7 gives integrability in every chart of this model; this gives the inclusion opposite to the upper inclusion of the theorem.
In the blowup coordinates $Z_1/Z_0=z$ and $Z_ j/Z_0=zt_ j$ for $j\geq 2$, the Jacobian of $\rho $ is $z^{q-1}$. Hence, if a section $s$ of $A^ m$ vanishes to order $d$ at $p$, then $T_ m(s)$ vanishes to order $d+q-1$ generically along $E$. The two divisorial thresholds $\lfloor m/2\rfloor $ and $K_ m$ therefore give exactly the orders $\ell _ m$ and $k_ m$ above. Homogeneous polynomials of degree $m$ realize every jet of total degree at most $m$ at $p$. For $m\geq 2q$ we have $k_ m=\ell _ m-1$, and the additional jets of degree $\ell _ m-1$ are spanned by $\binom {\ell _ m+q-2}{q-1}$ monomials, which gives the defect. For $m<2q$ both orders vanish.
Finally, in cohomology $[\beta ]=\rho ^*c_1(A)-\tfrac 12[E]$. The intersection numbers containing both $\rho ^*c_1(A)$ and $[E]$ vanish, because $\rho ^*c_1(A)$ restricts to zero on $E$. The normal bundle of $E\simeq \mathbb P^{q-1}$ is $\mathcal O_ E(-1)$, so $\int _ Y[E]^ q=(-1)^{q-1}$, while $\int _ Y(\rho ^*c_1(A))^ q=1$. Expanding gives $\int _ Y\beta ^ q=1-2^{-q}$, which is the mass of $\psi $ by the theorem.
$\square$
Pome
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