Proposition 22.1.98

Proposition 22.1.98.6A232EAE791C Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential and $\ell \in \mathcal{R}(X,\theta ;\phi )$ with $\sup _ X \ell _1\leq 0$. Let $\Phi $ be the complexification of $\ell $. Then the function

\[ v\mapsto -\sigma (v)(\Phi ) \quad \text{for } v\in X^{\operatorname{div}} \]

admits a unique extension to an element of $\mathrm{PSH}(L^{\mathrm{An}})$.

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

\[ \mathcal{I}(m\Phi )|_{X\times \Delta ^*} = p_1^* \mathcal{I}(m\phi )|_{X\times \Delta ^*}. \]

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

9186BF0A5F55 22.1.99

\begin{equation} \mathcal{I}(m\Phi )=\mathfrak {a}_0 +\mathfrak {a}_1 t+\cdots +\mathfrak {a}_{N-1} t^{N-1}+\mathfrak {a}_ N (t^ N)\, , \end{equation}

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

\[ \varphi _ m(v)\coloneqq -\frac{1}{m}\sigma (v)(\mathcal{I}(m\Phi ))=-\frac{1}{m}\min _ i (v(\mathfrak {a}_ i)+i),\quad v\in X^{\operatorname{div}}. \]

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

\[ -\sigma (v)(\Phi )=\lim _{m\to \infty }-\frac{1}{2^ m}\sigma (v)(\mathcal{I}(2^ m\Phi ))=\lim _{m\to \infty }\varphi _{2^ m}(v) \]

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. $\square$

Proposition 22.1.98`6A232EAE791C`

Comments

Proposition 22.1.986A232EAE791C

Comments

Proposition 22.1.98`6A232EAE791C`