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)...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-06-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.579/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206247050182656 |
|---|---|
| author | Boucksom, Sébastien Jonsson, Mattias |
| author_facet | Boucksom, Sébastien Jonsson, Mattias |
| author_sort | Boucksom, Sébastien |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-90d3608e002240a5834ea30747220f02 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-06-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-90d3608e002240a5834ea30747220f022025-02-07T11:13:30ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-06-01362S154210.5802/crmath.57910.5802/crmath.579Non-Archimedean Green’s functions and Zariski decompositionsBoucksom, Sébastien0Jonsson, Mattias1Sorbonne Université and Université Paris Cité, CNRS, IMJ-PRG, F-75005 Paris, FranceDept of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USAWe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.579/ |
| spellingShingle | Boucksom, Sébastien Jonsson, Mattias Non-Archimedean Green’s functions and Zariski decompositions Comptes Rendus. Mathématique |
| title | Non-Archimedean Green’s functions and Zariski decompositions |
| title_full | Non-Archimedean Green’s functions and Zariski decompositions |
| title_fullStr | Non-Archimedean Green’s functions and Zariski decompositions |
| title_full_unstemmed | Non-Archimedean Green’s functions and Zariski decompositions |
| title_short | Non-Archimedean Green’s functions and Zariski decompositions |
| title_sort | non archimedean green s functions and zariski decompositions |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.579/ |
| work_keys_str_mv | AT boucksomsebastien nonarchimedeangreensfunctionsandzariskidecompositions AT jonssonmattias nonarchimedeangreensfunctionsandzariskidecompositions |