Example 18.39.10.EB221EFD409A Let $B=\mathbb P^ q$ with a hyperplane $D$, let $A=\mathcal O_ B(1)$, and let $X=B\times \mathbb P^1$. Let $\omega _ B,\omega _ F$ be the Fubini–Study forms normalized to have total mass one, and use the corresponding metrics on the divisor line bundles. For a point $o\in \mathbb P^1$ set
\[ \varphi =\tfrac 14\log \bigl (\| s_ D\| ^4+\| s_ o\| ^6\bigr ),\qquad \omega =\omega _ B+\omega _ F. \]
Then $p=2$, $b_1=3$, $c=1/2$, $a=2/3$, and the reference potential $\psi =\tfrac 12\log \| s_ D\| ^2$ has mass $2^{-q}$. The spaces $V_1$ and $W_1$ agree, and for every $m\geq 2$ the defect is
\[ \dim V_ m-\dim W_ m =\binom {\lceil m/2\rceil +q}{q-1} \sim \frac{2^{1-q}}{(q-1)!}m^{q-1}. \]
For $q=1$ the defect is exactly one for every $m\geq 2$.
Proof.
Locally, the complex Hessian of the logarithm of a sum of two squared norms of holomorphic sections is bounded below by a convex combination of the negatives of their curvature forms. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -\tfrac 12\omega _ B-\tfrac 34\omega _ F$ off the pole set, and across it by the extension theorem after a quadratic correction. Thus $\varphi \in \mathrm{PSH}(X,\omega )$. The pole set is $D\times \{ o\} $, along which $\varphi $ has the exact local model of corollary 18.39.9. Here $u=0$, $\beta =\tfrac 12\omega _ B$, and $K_ m=L_ m-1$ for $m\geq 2$, while $K_1=L_1=0$. The spaces $V_ m$ and $W_ m$ therefore correspond to homogeneous polynomials of degrees $m-L_ m+1$ and $m-L_ m$, and since $m-L_ m=\lceil m/2\rceil $, the difference of their dimensions is the displayed binomial coefficient.
$\square$
Pome
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