Let be an ample line bundle on a smooth projective variety over a non-archimedean field . For a continuous metric on , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on 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 is a surface defined geometrically over the function field of a curve over a perfect field of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over . 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.
Soit un fibré en droites sur une variété projective lisse sur un corps non-archimédien . Pour une métrique continue sur , on montre dans les deux cas suivants que l’enveloppe semi-positive est une métrique continue semi-positive sur 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 est une surface définie géométriquement sur le corps de fonctions d’une courbe sur un corps parfait de caractéristique positive. Le deuxième cas reste valable en dimension supérieure sous l’hypothèse de ce que nous disposons de résolution de singularités sur . 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.
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Keywords: pluripotential theory, non-archimedean geometry, test ideals
Mot clés : théorie pluri-potentielle, géométrie non-archimédienne, idéaux test
Gubler, Walter 1; Jell, Philipp 2; Künnemann, Klaus 1; Martin, Florent 1
@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 = {https://aif.centre-mersenne.org/articles/10.5802/aif.3296/} }
TY - JOUR AU - Gubler, Walter AU - Jell, Philipp AU - Künnemann, Klaus AU - Martin, Florent TI - Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals JO - Annales de l'Institut Fourier PY - 2019 SP - 2331 EP - 2376 VL - 69 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3296/ DO - 10.5802/aif.3296 LA - en ID - AIF_2019__69_5_2331_0 ER -
%0 Journal Article %A Gubler, Walter %A Jell, Philipp %A Künnemann, Klaus %A Martin, Florent %T Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals %J Annales de l'Institut Fourier %D 2019 %P 2331-2376 %V 69 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3296/ %R 10.5802/aif.3296 %G en %F AIF_2019__69_5_2331_0
Gubler, Walter; Jell, Philipp; Künnemann, Klaus; Martin, Florent. Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals. Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2331-2376. doi : 10.5802/aif.3296. https://aif.centre-mersenne.org/articles/10.5802/aif.3296/
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