Corollary 18.39.9.094794C14195 Keep the positivity condition $\beta >0$ of theorem 18.39.8. Suppose that a nonempty compact smooth submanifold $Z\subset \pi ^{-1}(D)$ projects submersively onto $D$, and that near every point of $Z$ there are coordinates as in that theorem in which
\[ Z=\{ z=w_1=\cdots =w_ r=0\} ,\qquad \varphi =\frac cp\log \left(|z|^{2p}+\sum _{j=1}^ r|w_ j|^{2b_ j}\right)+O(1), \]
with fixed $c,p,b_ j$. Suppose also that $\varphi $ is locally bounded outside $Z$. Then $\varphi $ satisfies the hypotheses of theorem 18.39.8, so its conclusions hold, and moreover
\[ V_ m=H^0(B,A^ m(-K_ mD))\qquad (m\geq 1). \]
Proof.
The local model is bounded below by $c\log |z|^2$ up to a constant. Near points of $\pi ^{-1}(D)$ outside $Z$, the function $\varphi $ is locally bounded, so the same lower bound holds there; away from $\pi ^{-1}(D)$ both functions are locally bounded. By compactness, finitely many charts give the global lower bound. The upper bound holds in any chart of the model. Conversely, if a section is divisible by $z^{K_ m}$, its residual coefficient is bounded and holomorphic in each chart of the model, and the strict inequality $K_ m+1>mc-a$ together with lemma 18.39.7 gives integrability there. Outside $Z$ the function $\varphi $ is locally bounded, which proves the reverse inclusion.
$\square$
Pome
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