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22.1.70 Non-Archimedean plurisubharmonic functions3CE1C3E79114

Let $X$ be an irreducible complex projective variety of dimension $n$ and $L$ be a holomorphic pseudo-effective $\mathbb {Q}$-line bundle on $X$. Through the GAGA morphism $X^{\mathrm{An}}\rightarrow X$ of ringed spaces, $L$ can be pulled back to an analytic line bundle $L^{\mathrm{An}}$ on $X$. The purpose of this section is to study the psh metrics on $L^{\mathrm{An}}$. This description avoids the direct treatment of $L^{\mathrm{An}}$ itself.

We define $\mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}(L^{\mathrm{An}})$, the set of (rational) generically finite Fubini–Study functions $\phi \colon X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ that are of the following form:

13.37
\begin{equation} \phi =\frac{1}{m}\max _ j \{ \log |s_ j|+\lambda _ j\} . \end{equation}

Here $m\in \mathbb {Z}_{>0}$ is an integer such that $L^ m$ is a line bundle, the $s_ j$'s are a finite collection of non-zero sections in $\mathrm{H}^0(X,L^ m)$, and $\lambda _ j\in \mathbb {Q}$. We follow the convention of Boucksom–Jonsson by writing $\log |s_ j|(v)=-v(s_ j)$.

Definition 22.1.71 .EEADED50AAC3 A plurisubharmonic metric (or psh metric for short) on $L^{\mathrm{An}}$ is a function $\phi \colon X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ that is not identically $-\infty $, and is the pointwise limit of a decreasing net $(\phi _ i)_{i\in I}$, where $\phi _ i\in \mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}(L_ i^{\mathrm{An}})$ for some $\mathbb {Q}$-line bundles $L_ i$ on $X$ satisfying $c_1(L_ i)\to c_1(L)$ in $\mathrm{NS}^1(X)_{\mathbb {R}}$.

The set of psh metrics on $L^{\mathrm{An}}$ is denoted by $\mathrm{PSH}(L^{\mathrm{An}})$. We endow $\mathrm{PSH}(L^{\mathrm{An}})$ with the topology of pointwise convergence on $X^{\operatorname{div}}$. This topology is Hausdorff as functions in $\mathrm{PSH}(L^{\mathrm{An}})$ are completely determined by their restriction on $X^{\operatorname{div}}$:

Theorem 22.1.72 ([Theorem 4.22, BJ22]).0DAB63894803 Let $\phi \in \mathrm{PSH}(L^{\mathrm{An}})$ and $\psi :X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ be an usc function. Assume that $\phi \leq \psi $ on $X^{\operatorname{div}}$. Then the same holds on $X^{\mathrm{An}}$.

Proposition 22.1.74.85809FFB162F Let $H$ be an ample line bundle on $X$. Consider $\phi \in \mathrm{PSH}((L+H)^{\mathrm{An}})$. Assume that for each $m\in \mathbb {Z}_{>0}$, we have $\phi \in \mathrm{PSH}((L+m^{-1}H)^{\mathrm{An}})$. Then $\phi \in \mathrm{PSH}(L^{\mathrm{An}})$.

This is a special case of [(PSH2), BJ22] on Page 45. Next we note that we may use sequences instead of nets in the definition of $\mathrm{PSH}(L^{\mathrm{An}})$:

Theorem 22.1.75 ([Corollary 12.18, BJ22]).6B2CF0C2744A Let $S$ be an ample line bundle on $X$. Let $\phi \in \mathrm{PSH}(L^{\mathrm{An}})$. Then there is a sequence of rational numbers $\varepsilon _ i\searrow 0$ and a decreasing sequence $\phi _ i\in \mathcal{H}^{\mathrm{gf}}_{\mathbb {Q}}((L+\varepsilon _ i S)^{\mathrm{An}})$ such that $\phi $ is the pointwise limit of $\phi _ i$, as $i\to \infty $.

The space $\mathrm{PSH}(L^{\mathrm{An}})$ inherits most of the expected properties of (Archimedean) psh functions ([Theorem 4.7, BJ22]). However, the following compactness result is not known:

Conjecture 22.1.76 .95739E70234B Assume that $X$ is unibranch. Then every bounded from above increasing net of elements in $\mathrm{PSH}(L^{\mathrm{An}})$ converges in $\mathrm{PSH}(L^{\mathrm{An}})$.

This prediction is equivalent to the so-called envelope conjecture [Conjecture 5.14, BJ22]: the regularized supremum of a bounded from above family of functions in $\mathrm{PSH}(L^{\mathrm{An}})$ lies in $\mathrm{PSH}(L^{\mathrm{An}})$. See [Theorem 5.11, BJ22] for the proof of the equivalence. This conjecture is proved when $X$ is smooth and $L$ is nef in [BJ22]. More recently, in [BJ22b], Boucksom–Jonsson further established the case when $X$ is smooth and $L$ is pseudo-effective.
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