Example 18.39.20.C695A6659495 Let $q\geq 2$, let $\rho :B\to \mathbb P^ q$ be the blowup of $p=[1:0:\cdots :0]$, and let $E$ be its exceptional divisor. Let $\tau :B\to \mathbb P^{q-1}$ be the extension to $B$ of the projection from $p$, and put
\[ \alpha =\rho ^*\omega _{\mathbb P^ q},\qquad \gamma =\tau ^*\omega _{\mathbb P^{q-1}},\qquad \kappa =\alpha +\gamma >0, \]
where $\omega _{\mathbb P^ q}$ and $\omega _{\mathbb P^{q-1}}$ are the normalized Fubini–Study forms. Equip $\mathcal O_ B(E)=\rho ^*\mathcal O_{\mathbb P^ q}(1)\otimes \tau ^*\mathcal O_{\mathbb P^{q-1}}(-1)$ with the quotient metric, whose curvature is $\eta =\alpha -\gamma $, and let $s_ E$ be its canonical section. For an integer $k\geq 1$, equip
\[ A=\rho ^*\mathcal O_{\mathbb P^ q}(k)\otimes \mathcal O_ B(E) \quad \hbox{with curvature}\quad \theta =k\alpha +\eta . \]
Let $X=B\times \mathbb P^1$, with projection $\pi $ to $B$. For a point $o\in \mathbb P^1$, use the normalized Fubini–Study metric on $\mathcal O_{\mathbb P^1}(o)$, and set
\[ \begin{gathered} R=\| \pi ^*s_ E\| ^2,\quad T=\| s_ o\| ^2,\quad \varphi =\tfrac 12\log (R^2+T^3),\quad \omega =\pi ^*\kappa +2\omega _{\mathbb P^1},\\ \psi =\log \| s_ E\| ^2. \end{gathered} \]
Then $\varphi \in \mathrm{PSH}(X,\omega )$, $\psi \in \mathrm{PSH}(B,\theta )$ and
\[ \mathrm{MA}_\theta (\psi )=(k\alpha )^ q,\qquad \int _ B\mathrm{MA}_\theta (\psi )=k^ q. \]
The residual $k\alpha $ cannot be made strictly positive by a smooth change of gauge. Nevertheless, for every $m\geq 1$ the selected and reference spaces coincide as subspaces of $H^0(B,A^ m)$:
\[ V_ m=W_ m=H^0(B,A^ m) =s_ E^ m\rho ^*H^0(\mathbb P^ q,\mathcal O(km)), \qquad \dim V_ m=\binom {km+q}{q}. \]
Proof.
Since $(\rho ,\tau )$ is an embedding, $\kappa >0$. The quotient metric satisfies
\[ \| s_ E\| ^2=\rho ^*\left(\frac{\sum _{j=1}^ q|Z_ j|^2}{\sum _{j=0}^ q|Z_ j|^2}\right), \]
as in example 18.39.17. The same calculation over a common denominator, now with coefficient $1/2$, gives
\[ \mathrm{dd}^{\mathrm{c}}\varphi \geq -\pi ^*\alpha -\tfrac 32\omega _{\mathbb P^1} \]
off the pole set $E\times \{ o\} $, and across it by the extension theorem. Hence $\omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq \pi ^*\gamma +\tfrac 12\omega _{\mathbb P^1}\geq 0$. Along the smooth pole set $E\times \{ o\} $, the function $\varphi $ has the exact local model $\tfrac 12\log (|z|^4+|w|^6)+O(1)$, and it is locally bounded elsewhere. It also satisfies the global lower bound $\varphi \geq \log R$.
We may therefore take $c=1$, $u=0$, $p=2$, $b_1=3$ and $a=2/3$, with residual $\beta =\theta -\eta =k\alpha \geq 0$. Its top integral is $k^ q$, since the normalized Fubini–Study form has top integral one and $\rho $ is a modification. Corollary 18.39.19, Corollary 18.39.9 therefore give
\[ W_ m=H^0(B,A^ m(-mE)),\qquad V_ m=H^0(B,A^ m(-(m-1)E)). \]
Let $t\geq 0$ be an integer. A section of $\rho ^*\mathcal O(km)\otimes \mathcal O_ B(tE)$, divided by $s_ E^ t$, gives a holomorphic section of $\mathcal O(km)$ on $\mathbb P^ q\setminus \{ p\} $, which extends across $p$ by Hartogs' theorem since $q\geq 2$. Pulling the extension back and multiplying by $s_ E^ t$ recovers the original section off $E$, hence everywhere by the identity theorem. Thus every section of $A^ m$ has the form $s_ E^ m\rho ^*s$, and the natural embeddings of $W_ m$ and $V_ m$ into $H^0(B,A^ m)$ both have this image. Its dimension is the number $\binom {km+q}{q}$ of monomials of degree $km$.
Finally, let $C\subset E\simeq \mathbb P^{q-1}$ be a projective line; it is contracted by $\rho $. For every smooth real function $u$, Stokes' theorem gives $\int _ C(k\alpha +\mathrm{dd}^{\mathrm{c}}u)=0$, whereas a strictly positive form has positive integral over $C$. Changing the smooth metric on the divisor bundle changes the residual only by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, so this proves the obstruction. Moreover, $\varphi $ is finite on most of each fiber over $E$ and equals $-\infty $ at the point $o$ of that fiber, so it does not descend, even up to a bounded function.
$\square$
Pome
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