Non-Archimedean Green’s functions and Zariski decompositions

Boucksom, Sébastien; Jonsson, Mattias

Comptes Rendus. Mathématique, Complex algebraic geometry, in memory of Jean-Pierre Demailly, Tome 362 (2024), pp. 5-42

We study the non-Archimedean Monge–Ampère 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 $\mathbb{Q}$-PL (i.e. not a model function in the discretely valued case). Second, we produce an example of a function whose Monge–Ampère 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.

Nous étudions l’équation de Monge–Ampère non-archimédienne sur une variété projective lisse sur un corps de valuation discrète ou triviale. Tout d’abord, nous donnons un exemple de fonction de Green, associée à une valuation divisorielle, qui n’est pas $\mathbb{Q}$-PL (i.e. pas une fonction modèle, dans le cas de valuation discrète). Ensuite, nous produisons un exemple de fonction dont la mesure de Monge–Ampère est à support dans un complexe dual, mais qui n’est invariante par la rétraction associée à aucun modele snc. Ceci répond négativement à une question de Burgos Gil et al. Nos exemples sont basés sur des constructions géométriques de Cutkosky et Lesieutre, et sont produits par changement de base à partir de fonctions de Green sur un corps trivialement valué ; cette théorie nous permet d’encoder de façon efficace la décomposition de Zariski de toute classe pseudo-effective.

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