Theorem 6.12.85BA222439952

Proof. Let $p\in \pi ^{-1}(V)$, and choose coordinates $w$ at $p$ as before (6.12.7), so that $E_1,\ldots ,E_\ell $ are the components of $D$ through $p$ and $E_ i=\{ w_ i=0\} $ near $p$. Since $b$ is bounded, (6.12.7), applied to $\pi \colon Y\to V$, gives

near $p$.

Let $h$ be holomorphic near $p$; we decide when $|h|^2\prod _{i\leq \ell }|w_ i|^{-2cd_ i}$, or equivalently $|h|^2\mathrm e^{-\varphi \circ \pi }$, is integrable near $p$. Let $k_ i$ be the order of $h$ along $E_ i$ at $p$, which is finite if $h$ does not vanish identically near $p$; if it does, the integrability is trivial and $k_ i=+\infty $. We claim that the integrability holds if and only if $k_ i+1>cd_ i$ for $i=1,\ldots ,\ell $. If these inequalities hold, write $h=\prod _ iw_ i^{k_ i}\tilde h$ with $\tilde h$ holomorphic, by lemma 6.12.2; the integrand is at most a constant times $\prod _ i|w_ i|^{2k_ i-2cd_ i}$, which is integrable near $p$ by Fubini's theorem and polar coordinates in each $w_ i$, since $2k_ i-2cd_ i>-2$. Conversely, suppose that $k_1+1\leq cd_1$, say, and write $h=w_1^{k_1}\tilde h$ with $\tilde h$ not divisible by $w_1$. Then $\tilde h$ does not vanish identically on $\{ w_1=0\} $ near $p$, by lemma 6.12.1(2), so arbitrarily close to $p$ there are an open set $W'$ of points $w'=(w_2,\ldots ,w_ n)$ and $\delta ,\rho >0$ with $|\tilde h(w_1,w')|\geq \delta $ for $|w_1|<\rho $ and $w'\in W'$. There the integrand is at least a positive constant times $|w_1|^{2k_1-2cd_1}$, since the other factors $|w_ i|^{-2cd_ i}$ are bounded below; as $2k_1-2cd_1\leq -2$, its integral in $w_1$ diverges, and by Fubini's theorem the integrand is not integrable near $p$.

Now let $f$ be holomorphic on $V$, and apply this to $h=(f\circ \pi )J_\pi $, with $J_\pi $ computed in local coordinates near $p$. The function $f\circ \pi $ is holomorphic on $\pi ^{-1}(V)$, which contains $E_ i$, so its order along $E_ i$ at $p$ is $\operatorname {ord}_{E_ i}f$, by lemma 6.12.1(3) and definition 6.12.3; the order of $J_\pi $ along $E_ i$ at $p$ is $\operatorname {ord}_{E_ i}J_\pi $. Since orders add under products (lemma 6.12.1(4)), $k_ i=\operatorname {ord}_{E_ i}f+\operatorname {ord}_{E_ i}J_\pi =\operatorname {ord}_{E_ i}f+A_ X(E_ i)-1$. Thus, for $p\in \pi ^{-1}(x)$, the condition at $p$ is (6.12.9) for the components through $p$. As $p$ ranges over $\pi ^{-1}(x)$, these are exactly the components meeting the fiber, and proposition 6.12.4 proves the first assertion.

For (6.12.10), let $U\subset V$ be open and $f\in \mathcal O(U)$, and put $h=(f\circ \pi )J_\pi $ on $\pi ^{-1}(U)$. For an integer $k$, the strict inequality $k+1>cd_ i$ is equivalent to $k\geq \lfloor cd_ i\rfloor $. By proposition 6.12.4 and the claim, applied at every point $p\in \pi ^{-1}(U)$, we have $f\in \mathcal I(\varphi )(U)$ if and only if at every such $p$ the order of $h$ at $p$ along each component $E_ i$ through $p$ is at least $\lfloor cd_ i\rfloor $. By lemma 6.12.2, this means that $h$ is divisible near every point of $\pi ^{-1}(U)$ by the local equation $\prod _ iw_ i^{\lfloor cd_ i\rfloor }$ of $\lfloor cD\rfloor $, that is, that $f\circ \pi =h/J_\pi $ is a section of $\mathcal O_ Y(K_{Y/X}-\lfloor cD\rfloor )$ over $\pi ^{-1}(U)$. Since $\lfloor cD\rfloor $ is effective, every section of $\mathcal O_ Y(K_{Y/X}-\lfloor cD\rfloor )$ over $\pi ^{-1}(U)$ is a section of $\mathcal O_ Y(K_{Y/X})$, and hence of the form $f\circ \pi $ with $f\in \mathcal O(U)$, by proposition 6.12.4. This proves (6.12.10) on $V$. $\square$