Non-Archimedean Green's functions and Zariski decompositions

Sebastien Boucksom; Mattias Jonsson

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arXiv · 2304.00144

Non-Archimedean Green's functions and Zariski decompositions

Sebastien Boucksom·Mattias Jonsson

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Abstract

We study the non-Archimedean Monge-Amp\`ere equation on a smooth projective variety over a discretely or trivially valued field. First, we give an example of a Green's function, associated to a divisorial valuation, which is not Q-PL (i.e. not a model function in the discretely valued case). Second, we produce an example of a function whose Monge-Amp\`ere measure is a finite atomic measure supported in a dual complex, but which is not invariant under the retraction associated to any snc model. This answers a question by Burgos Gil et al in the negative. Our examples are based on geometric constructions by Cutkosky and Lesieutre, and arise via base change from Green's functions over a trivially valued field; this theory allows us to efficiently encode the Zariski decomposition of a pseudoeffective numerical class.

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Sebastien Boucksom, Mattias Jonsson. 2023-03-31. Non-Archimedean Green's functions and Zariski decompositions. https://arxiv.org/abs/2304.00144

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