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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
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