Non-Archimedean Green’s functions and Zariski decompositions
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)...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-06-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.579/ |
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| Summary: | 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. |
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| ISSN: | 1778-3569 |