## Singular semipositive metrics in non-Archimedean geometry.(English)Zbl 1346.14065

Let $$k$$ be a field of characteristic 0 and let $$K=k((t))$$ be the field of Laurent series over it. Let $$X$$ be the analytification, in the sense of Berkovich theory, of a smooth projective variety over $$K$$. Let $$L$$ be an ample line bundle on $$X$$.
In [J. Algebr. Geom. 4, No. 2, 281–300 (1995; Zbl 0861.14019)], S.-W. Zhang introduced a notion of positivity for metrics on $$L$$. By a classical construction, to any model $$\mathcal{X}$$ of $$X$$ over $$R=k[[T]]$$ and any model $$\mathcal{L}$$ of $$L$$ over $$\mathcal{X}$$, one can associate a continuous metric $$\| \cdot\|_{\mathcal{L}}$$ on $$X$$. When $$\mathcal{L}$$ is nef on the special fiber of $$\mathcal{X}$$, this metric is called a semipositive model metric. In general, a continuous metric on $$L$$ is called semipositive when it is a uniform limit of semipositive model metrics.
The aim of the present article is to extend the notion of positivity to non-necessarily continuous metrics and to prove that it satisfies the expected basic properties. To do so, the authors use fine topological properties of $$X$$: it is homeomorphic to the projective limit $$\varprojlim_{\mathcal{X}} \Delta_{\mathcal{X}}$$, where $$\mathcal{X}$$ runs through the set of SNC models of $$X$$ (regular models whose special fiber is simple normal crossing) and $$\Delta_{\mathcal{X}}$$ is a compact simplicial complex. More precisely, $$\Delta_{\mathcal{X}}$$ may be realized as the dual complex of the special fiber of $$\mathcal{X}$$ (encoding the multiple intersections between its irreducible components), it embeds canonically into $$X$$ and there is a retraction $$p_{\mathcal{X}}$$ on $$X$$ onto its image (still denoted by $$\Delta_{\mathcal{X}}$$). The fact that only SNC models need to be considered follows from desingularization results by M. Temkin (see [Adv. Math. 219, No. 2, 488–522 (2008; Zbl 1146.14009)]) and use the fact that the residue field $$k$$ of $$K$$ has characteristic 0.
From now on we fix a reference model metric $$\|\cdot\|$$ with curvature form $$\theta$$. The authors say that a function $$\varphi$$ on $$X$$ is a $$\theta$$-plurisubharmonic ($$\theta$$-psh) model function when $$\|\cdot\| e^{-\varphi}$$ is a semipositive model metric. A general $$\theta$$-psh function on $$X$$ is then defined to be an upper semicontinuous function such that, for each SNC model $$\mathcal{X}$$ of $$X$$, we have $$\varphi \leq \varphi \circ p_{\mathcal{X}}$$ and the restriction of $$\varphi$$ to $$X$$ is a uniform limit of restrictions of $$\theta$$-psh model functions.
The main results of the paper are analogues of well-known results in the complex case: A) the set of $$\theta$$-psh functions on $$X$$ moduling scaling is compact; B) every $$\theta$$-psh function is the pointwise limit of a decreasing net of $$\theta$$-psh model functions. The proofs use computations of intersection numbers on the special fibers of the models, toroidal techniques to construct appropriate models and a certain cohomological vanishing property of multiplier ideals (that also requires residue characteristic 0).

### MSC:

 14G22 Rigid analytic geometry 32U05 Plurisubharmonic functions and generalizations

### Citations:

Zbl 0861.14019; Zbl 1146.14009
Full Text:

### References:

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