22.1.91 The algebraic settingF432DF631582

Proof. After replacing $L$ with a sufficiently high power, we may assume that $L$ is a line bundle. Take a very ample line bundle $H$ on $X$. By Siu's uniform global generation theorem [Siu98], [Theorem 6.27, Dem12] there exists $b>0$ large enough so that $H^{b}\otimes L^ k\otimes \mathcal{I}(k\varphi )$ is globally generated for all $k>0$. Let $\{ s_ i\} _ i$ be a finite set of global sections that generate the sheaf $H^{b}\otimes L^ k\otimes \mathcal{I}(k\varphi )$. Then

It follows that $v\mapsto -k^{-1}v\left(\mathcal{I}(k\varphi ) \right)$ lies in $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+\frac{b}{k}H)^{\mathrm{An}})$. Using (22.1.81), we conclude that $\varphi ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$. $\square$

Proof. The result follows from (22.1.87), proposition 22.1.73 and 22.1.92. $\square$

Proof. It suffices to argue that $I$ can be expanded as in (22.1.95). To see this, assume that $a\in I$. We can write $a=a_0+a_1t+\cdots +a_ mt^ m$ with $a_ i\in R$. Then our assumption implies that $\sum _ i a_ i \rho ^ i t^ i\in I$ as well for all $\rho \in S^1$. So by the Lagrange interpolation formula, $a_ it^ i\in I$ for all $i$. Therefore, we can write $I$ as $I_0+I_1t+I_2t^2+\cdots $ for some ideals $I_0\subseteq I_1\subseteq \cdots $ in $R$. But as $R$ is noetherian, there is $m\geq 0$ so that $I_{m'}=I_ m$ for $m'>m$. (22.1.95) follows. $\square$

Proof. Let $q:X\times (\mathbb {P}^1\setminus \{ 0\} )\rightarrow X$ be the natural projection. Since $\mathbb C^* \subset \mathbb {P}^1\setminus \{ 0\} $, we can glue $q^*\mathcal{J}$ with $\mathcal{I}$ to get an analytic coherent ideal sheaf on $X\times \mathbb {P}^1$. By the GAGA principle, this ideal sheaf is necessarily algebraic, hence so is its restriction to $X\times \mathbb {C}$. $\square$

Proof. The argument for this is exactly the same as the one in [Lemma 5.6, BBJ21] with Nadel's vanishing theorem replaced by the family version proved by Matsumura in [Theorem 1.7, Mat16]. $\square$

Proof. We may assume that $L$ is a line bundle. Observe that the extension is unique if it exists by theorem 22.1.72.

Let $p_1\colon X\times \mathbb {C}\rightarrow X$ be the projection. For each $t>0$, we have $\ell _ t\in \mathcal{E}(X,\theta ;\phi )$, hence $\ell _ t\sim _ P\phi $ by theorem 16.3.15 and $\ell _ t\sim _{\mathcal{I}}\phi $ by corollary 16.8.14. Thanks to proposition 5.16.22 and 20.2.56, for each $m\in \mathbb {Z}_{>0}$, we have

In particular, $\mathcal{I}(m\Phi )$ admits an $S^1$-invariant coherent extension to $X\times \mathbb {C}$, obtained by gluing it with $p_1^*\mathcal{I}(m\phi )$ over $X\times \mathbb {C}^*$.

From 22.1.94 and 22.1.96, we get that

where the $\mathfrak {a}_ i$'s are coherent ideal sheaves on $X$.

Using 22.1.97, there exists an ample line bundle $T$ over $X$ such that $p_1^* (T\otimes L^ m)\otimes \mathcal{I}(m\Phi )$ is globally generated, which is equivalent to $T\otimes L^ m\otimes \mathfrak {a}_ i$ being globally generated for all $i$.⁴

We define

From the right-hand side of the formula, $\varphi _ m$ can be extended to an element in $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+m^{-1}T)^{\mathrm{An}})$, which we denote by the same symbol.

For $v\in X^{\operatorname{div}}$,

and the right-hand side defines an element in $\mathrm{PSH}(L^{\mathrm{An}})$ by definition, since $\{ \varphi _{2^ m}\} _ m$ is decreasing by the subadditivity of multiplier ideals theorem 5.16.14. $\square$

Proof. Observe that the extension is unique if it exists by theorem 22.1.72. We may assume that $\Gamma _{\max }=0$ without loss of generality.

When $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, our assertion follows from proposition 22.1.98 and proposition 22.1.84.

In general, fix an ample line bundle $H$ on $X$ and a Kähler form $\omega \in c_1(H)$. Then we know that

for any $m\in \mathbb {Z}_{>0}$. Therefore, $\Gamma ^{\mathrm{An}}\in \mathrm{PSH}(L^{\mathrm{An}})$ by proposition 22.1.74. $\square$