Theorem 18.39.16.1FC6712A00F0 Suppose that $c_ i>0$ and $u\in C^\infty (Y,\mathbb R)$ satisfy
\[ \beta =\theta '+\mathrm{dd}^{\mathrm{c}}u-\sum _ i c_ i\eta _ i>0, \qquad \varphi '\geq \sum _ i c_ i\log \| \pi '^*s_ i\| _{H_ i}^2-C. \]
Suppose that for each $i$ the function $\varphi '$ satisfies an anisotropic upper bound, with respect to $\pi '$, on a submersion polydisk over an open subset of $D_ i$ away from the other divisors, as in theorem 18.39.13, with fixed $p_ i,b_{ij}$. Define $a_ i$, $L_{m,i}$ and $K_{m,i}$ by the formulas of that theorem, and put
\[ \psi '=u+\sum _ i c_ i\log \| s_ i\| _{H_ i}^2, \qquad M_ m=\rho ^*A^ m\otimes R_\rho . \]
Then there is a unique $\psi \in \mathrm{PSH}(B,\theta )$ with $\psi \circ \rho =\psi '$, and
\[ \mathrm{MA}_\theta (\psi )=\rho _*(\beta ^ q),\qquad \int _ B\mathrm{MA}_\theta (\psi )=\int _ Y\beta ^ q\in (0,\infty ). \]
The map $T_ m(s)=J_\rho \rho ^*s$ identifies $H^0(B,A^ m)$ with $H^0(Y,M_ m)$, and for $W_ m=H^0(B,A^ m\otimes \mathcal I(m\psi ))$,
\[ \begin{split} T_ m(W_ m)& =H^0\left(Y,M_ m\left(-\sum _ iL_{m,i}D_ i\right)\right),\\ T_ m(W_ m)& \subseteq T_ m(V_ m) \subseteq H^0\left(Y,M_ m\left(-\sum _ iK_{m,i}D_ i\right)\right). \end{split} \]
Consequently $W_ m\subseteq V_ m$ and $\dim V_ m-\dim W_ m=O(m^{q-1})$. If $q=1$, the difference is at most $\sum _ i\lceil a_ i\rceil $ for every $m$.
Proof.
By the Poincaré–Lelong formula, $\theta '+\mathrm{dd}^{\mathrm{c}}\psi '=\beta +\sum _ i c_ i[D_ i]$. Thus $\psi '\in \mathrm{PSH}(Y,\rho ^*\theta )$, and proposition 5.43.18 gives the unique potential $\psi $ on $B$ with $\psi \circ \rho =\psi '$. On $Y$, locality off the union of the divisors and the fact that the non-pluripolar measure does not charge this union give $\mathrm{MA}_{\theta '}(\psi ')=\beta ^ q$. The measure identity in proposition 18.35.1 then gives the pushforward formula and the mass.
Lemma 5.43.22, applied first with weight zero and then with weight $m\psi $, gives the identification of the full section spaces and the formula for $T_ m(W_ m)$; here $\mathcal I(m\psi ')=\mathcal O_ Y(-\sum _ iL_{m,i}D_ i)$ by the integrability computation for simple normal crossings in the proof of theorem 18.39.13. Since $u$ is bounded, the global lower bound gives $\varphi '\geq \psi '\circ \pi '-C'$. As $\rho \circ \pi '=\pi \circ \mu $ and $\mu $ is surjective, this descends to $\varphi \geq \psi \circ \pi -C'$, and integrability in bounded submersion charts proves $W_ m\subseteq V_ m$.
For $s\in V_ m$, the transformation of multiplier ideals under $\mu $ gives
\[ J_\mu \mu ^*s=\pi '^*T_ m(s) \in H^0\bigl (X',\pi '^*M_ m\otimes \mathcal I(m\varphi ')\bigr ). \]
Hence, in each polydisk where the upper bound holds, the descended coefficient of $T_ m(s)$ vanishes to order at least $K_{m,i}$ along $D_ i$, by lemma 18.39.7 and the argument with a unit coefficient in the proof of theorem 18.39.8. This is the vanishing order of $T_ m(s)$, including the Jacobian factor, and it proves the upper inclusion.
The chain of restrictions along the divisors on $Y$ has at most $\sum _ i\lceil a_ i\rceil $ steps. On $D_ i$, each target is $(\rho ^*A|_{D_ i})^ m$ twisted by powers of the bundles $\mathcal O_ Y(-D_ j)|_{D_ i}$ in a linear range and by the fixed bundle $R_\rho |_{D_ i}$. Including the latter in lemma 18.39.12 with exponent one, we see that every target has dimension $O(m^{q-1})$, uniformly. Summing gives the estimate. For $q=1$ each divisor is a point and every target has dimension one.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.