Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals
[Continuité des enveloppes plurisousharmoniques en géométrie non-archimédienne et idéaux test]
Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2331-2376.

Soit L un fibré en droites sur une variété projective lisse sur un corps non-archimédien K. Pour une métrique continue sur L an , on montre dans les deux cas suivants que l’enveloppe semi-positive est une métrique continue semi-positive sur L an et que l’équation de Monge–Ampère non-archimédienne a une solution. On le montre dans le premier cas pour les courbes en utilisant des résultats de Thuillier. Dans un deuxième cas, on le montre quand X est une surface définie géométriquement sur le corps de fonctions d’une courbe sur un corps parfait k de caractéristique positive. Le deuxième cas reste valable en dimension supérieure sous l’hypothèse de ce que nous disposons de résolution de singularités sur k. La preuve suit une stratégie de Boucksom, Favre et Jonsson, en remplaçant les idéaux multiplicateurs par des idéaux test. Finalement, l’appendice de Burgos et Sombra fournit un exemple d’une métrique semi-positive dont la rétraction n’est pas semi-positive. L’exemple est basé sur la construction d’une variété torique qui a deux modèles SNC qui induisent le même squelette mais des applications de rétraction différentes.

Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L an , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L an and that the non-archimedean Monge–Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.

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DOI : https://doi.org/10.5802/aif.3296
Classification : 32P05,  13A35,  14G22,  32U05
Mots clés : théorie pluri-potentielle, géométrie non-archimédienne, idéaux test
@article{AIF_2019__69_5_2331_0,
     author = {Gubler, Walter and Jell, Philipp and K\"unnemann, Klaus and Martin, Florent},
     title = {Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals},
     journal = {Annales de l'Institut Fourier},
     pages = {2331--2376},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {5},
     year = {2019},
     doi = {10.5802/aif.3296},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3296/}
}
Gubler, Walter; Jell, Philipp; Künnemann, Klaus; Martin, Florent. Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals. Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 2331-2376. doi : 10.5802/aif.3296. http://archive.numdam.org/articles/10.5802/aif.3296/

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