Example 18.39.14.411D2D5FFC5D Let $B=(\mathbb P^1)^ q$, $X=B\times (\mathbb P^1)^ q$ and $A=\mathcal O_ B(1,\ldots ,1)$. For each $i$, let $D_ i\subset B$ be the inverse image of a marked point in the $i$th factor of $B$, and let $E_ i\subset X$ be the inverse image of a marked point in the $i$th factor of the fiber. Use normalized Fubini–Study metrics on the corresponding line bundles, and write $\omega _{B,i},\omega _{F,i}$ for the pullbacks to $X$ of the normalized Fubini–Study forms on the $i$th factors of $B$ and of the fiber. Put
\[ \begin{gathered} R_ i=\| \pi ^*s_ i\| ^2,\quad T_ i=\| s_{E_ i}\| ^2,\quad \varphi =\frac14\sum _{i=1}^ q\log (R_ i^2+T_ i^3),\\ \omega =\sum _ i(\omega _{B,i}+\omega _{F,i}),\qquad \psi =\frac12\sum _ i\log \| s_ i\| ^2. \end{gathered} \]
Then $\varphi \in \mathrm{PSH}(X,\omega )$, and theorem 18.39.13 applies with $u=0$, $c_ i=1/2$, $p_ i=2$, $b_{i1}=3$ and $a_ i=2/3$. The reference potential $\psi $ has mass $q!/2^ q$. With $L_ m=\lfloor m/2\rfloor $ and $K_ m=\max \{ \lfloor m/2-2/3\rfloor ,0\} $, the selected and reference spaces are exactly
\[ \begin{split} V_ m& =H^0\bigl (B,\mathcal O_ B(m-K_ m,\ldots ,m-K_ m)\bigr ),\\ W_ m& =H^0\bigl (B,\mathcal O_ B(m-L_ m,\ldots ,m-L_ m)\bigr ). \end{split} \]
They agree for $m=1$, and for $m\geq 2$ the defect is
\[ \dim V_ m-\dim W_ m =(\lceil m/2\rceil +2)^ q-(\lceil m/2\rceil +1)^ q. \]
For $q\geq 2$ this is $q2^{1-q}m^{q-1}+O(m^{q-2})$, and for $q=1$ it is exactly one.
Proof.
By the calculation in example 18.39.10, the complex Hessian of each summand of $\varphi $ is bounded below by $-\tfrac 12\omega _{B,i}-\tfrac 34\omega _{F,i}$, also across its pole set by the extension theorem after a quadratic correction. Summing proves $\varphi \in \mathrm{PSH}(X,\omega )$. The global lower bound follows from $R_ i^2+T_ i^3\geq R_ i^2$. For the upper bound over $D_ i$, away from the other divisors, place the $i$th fiber coordinate at its marked point; all the other summands are smooth there. The residual is $\beta =\tfrac 12\sum _ i\omega _{B,i}>0$, whose top integral is $q!/2^ q$.
The theorem gives the required generic orders along every $D_ i$. Conversely, near a point where several of the pairs $(R_ i,T_ i)$ vanish, the density $\mathrm{e}^{-m\varphi }$ is comparable to the product of the factors $(|z_ i|^4+|w_ i|^6)^{-m/4}$ in disjoint pairs of variables. A descended coefficient divisible by every $z_ i^{K_ m}$ has a bounded holomorphic residual. Each factor is integrable because $K_ m+1>m/2-2/3$, and Fubini's theorem gives integrability at the intersections; elsewhere the same argument uses fewer factors. This proves the identity for $V_ m$, and the theorem gives $W_ m$. A line bundle $\mathcal O_ B(d,\ldots ,d)$ with $d\geq 0$ has $(d+1)^ q$ independent sections, the products of monomials in the separate factors. Since $K_ m=L_ m-1$ for $m\geq 2$ and $m-L_ m=\lceil m/2\rceil $, subtraction gives the displayed defect and its leading term.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.