Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals
Annales de l'Institut Fourier, Volume 69 (2019) no. 5, pp. 2331-2376.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3296
Classification: 32P05,  13A35,  14G22,  32U05
Keywords: pluripotential theory, non-archimedean geometry, test ideals
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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/

[1] Abbes, Ahmed Éléments de géométrie rigide. Vol. I. Construction et étude géométrique des espaces rigides., Progress in Mathematics, Tome 286, Birkhäuser, 2011 | Article | Zbl: 1223.14003

[2] Baker, Matthew; Payne, Sam; Rabinoff, Joseph On the structure of non-Archimedean analytic curves, Tropical and non-Archimedean geometry (Contemporary Mathematics) Tome 605, American Mathematical Society, 2013, pp. 93-121 | Article | MR: 3204269 | Zbl: 1320.14040

[3] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, Tome 33, American Mathematical Society, 1990, x+169 pages | MR: 1070709 | Zbl: 0715.14013

[4] Berkovich, Vladimir G. Etale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Tome 78 (1993) no. 1, pp. 5-161 | Article | Zbl: 0804.32019

[5] Berkovich, Vladimir G. Smooth p-adic analytic spaces are locally contractible, Invent. Math., Tome 137 (1999) no. 1, pp. 1-84 | Article | MR: 1702143 | Zbl: 0930.32016

[6] Blickle, Manuel; Mustaţă, Mircea; Smith, Karen E. Discreteness and rationality of F-thresholds, Mich. Math. J., Tome 57 (2008), pp. 43-61 (Special volume in honor of Melvin Hochster) | Article | MR: 2492440 | Zbl: 1177.13013

[7] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, Tome 261, Springer, 1984, xii+436 pages | MR: 746961 | Zbl: 0539.14017

[8] Bosch, Siegfried; Lütkebohmert, Werner Stable reduction and uniformization of abelian varieties. I, Math. Ann., Tome 270 (1985) no. 3, pp. 349-379 | Article | MR: 774362 | Zbl: 0554.14012

[9] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Solution to a non-Archimedean Monge-Ampère equation, J. Am. Math. Soc., Tome 28 (2015) no. 3, pp. 617-667 | Article | MR: 3327532 | Zbl: 1325.32021

[10] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias The non-Archimedean Monge–Ampère equation., Nonarchimedean and tropical geometry. Based on two Simons symposia, Island of St. John, March 31 – April 6, 2013 and Puerto Rico, February 1–7, 2015 (Simons Symposia), Springer, 2016, pp. 31-49 | Article | Zbl: 1360.32018

[11] Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias Singular semipositive metrics in non-Archimedean geometry, J. Algebr. Geom., Tome 25 (2016) no. 1, pp. 77-139 | MR: 3419957 | Zbl: 1346.14065

[12] Brodmann, Markus P.; Sharp, Rodney Y. Local cohomology, Cambridge Studies in Advanced Mathematics, Tome 136, Cambridge University Press, 2013, xxii+491 pages | MR: 3014449 | Zbl: 1044.13007

[13] Burgos Gil, José Ignacio; Gubler, Walter; Jell, Philipp; Künnemann, Klaus; Martin, Florent Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld) (2016) (http://arxiv.org/abs/1608.01919)

[14] Burgos Gil, José Ignacio; Moriwaki, Atsushi; Philippon, Patrice; Sombra, Martín Arithmetic positivity on toric varieties, J. Algebr. Geom., Tome 25 (2016) no. 2, pp. 201-272 | Article | MR: 3466351 | Zbl: 1378.14048

[15] Burgos Gil, José Ignacio; Philippon, Patrice; Sombra, Martín Arithmetic geometry of toric varieties. Metrics, measures and heights, Astérisque, Tome 360, Société Mathématique de France, 2014 | Zbl: 1311.14050

[16] Calabi, Eugenio The space of Kähler metrics, Proceedings of the International Congress of Mathematicians (Amsterdam, 1954). Vol. 2 (1954), p. 206-207

[17] Calabi, Eugenio On Kähler manifolds with vanishing canonical class, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, 1957, pp. 78-89 | Article | MR: 0085583 | Zbl: 0080.15002

[18] Chambert-Loir, Antoine Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math., Tome 595 (2006), pp. 215-235 | Article | MR: 2244803 | Zbl: 1112.14022

[19] Cossart, Vincent; Piltant, Olivier Resolution of singularities of threefolds in positive characteristic. I., J. Algebra, Tome 320 (2008) no. 3, pp. 1051-1082 | Article | MR: 2427629 | Zbl: 1159.14009

[20] Cossart, Vincent; Piltant, Olivier Resolution of singularities of threefolds in positive characteristic. II, J. Algebra, Tome 321 (2009) no. 7, pp. 1836-1976 | Article | MR: 2494751 | Zbl: 1173.14012

[21] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of line bundles, Pure Appl. Math. Q., Tome 1 (2005) no. 2, pp. 379-403 | Article | MR: 2194730 | Zbl: 1127.14010

[22] Freitag, Eberhard; Kiehl, Reinhardt Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 13, Springer, 1988, xviii+317 pages | MR: 926276 | Zbl: 0643.14012

[23] Görtz, Ulrich; Wedhorn, Torsten Algebraic geometry I. Schemes, Advanced Lectures in Mathematics, Vieweg+Teubner, 2010, viii+615 pages | Article | MR: 2675155 | Zbl: 1213.14001

[24] Grothendieck, Alexander; Dieudonné, Jean Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents, Publ. Math., Inst. Hautes Étud. Sci., Tome 11, 17 (1961-63)

[25] Grothendieck, Alexander; Dieudonné, Jean Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, Publ. Math., Inst. Hautes Étud. Sci., Tome 20, 24, 28, 32 (1964–67)

[26] Gubler, Walter Local heights of subvarieties over non-Archimedean fields, J. Reine Angew. Math., Tome 498 (1998), pp. 61-113 | Article | MR: 1629925 | Zbl: 0906.14013

[27] Gubler, Walter; Hertel, Julius Local heights of toric varieties over non-archimedean fields, Actes de la Conférence “Non-Archimedean Analytic Geometry: Theory and Practice” (Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres) Tome 2017/1, Presses Universitaires de Franche-Comté, 2017, pp. 5-77 | MR: 3752487 | Zbl: 06854884

[28] Gubler, Walter; Martin, Florent On Zhang’s semipositive metrics, Doc. Math., Tome 24 (2019), pp. 331-372 | MR: 3960125 | Zbl: 07050188

[29] Gubler, Walter; Rabinoff, Joseph; Werner, Annette Skeletons and tropicalizations, Adv. Math., Tome 294 (2016), pp. 150-215 | Article | MR: 3479562 | Zbl: 1370.14024

[30] Hara, Nobuo; Yoshida, Ken-Ichi A generalization of tight closure and multiplier ideals, Trans. Am. Math. Soc., Tome 355 (2003) no. 8, pp. 3143-3174 | Article | MR: 1974679 | Zbl: 1028.13003

[31] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, Tome 52, Springer, 1977, xvi+496 pages | MR: 0463157 | Zbl: 0367.14001

[32] Jell, Philipp Differential forms on Berkovich analytic spaces and their cohomology (2016) (Ph. D. Thesis)

[33] Jonsson, Mattias Dynamics of Berkovich spaces in low dimensions, Berkovich spaces and applications (Lecture Notes in Mathematics) Tome 2119, Springer, 2015, pp. 205-366 | Article | MR: 3330767 | Zbl: 1401.37103

[34] Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David Uniform bounds for the number of rational points on curves of small Mordell-Weil rank, Duke Math. J., Tome 165 (2016) no. 16, pp. 3189-3240 | Article | MR: 3566201 | Zbl: 06666955

[35] Katz, Nicholas M. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Publ. Math., Inst. Hautes Étud. Sci., Tome 39 (1970), pp. 175-232 | Article | MR: 0291177 | Zbl: 0221.14007

[36] Keeler, Dennis S. Ample filters of invertible sheaves, J. Algebra, Tome 259 (2003) no. 1, pp. 243-283 | Article | MR: 1953719 | Zbl: 1082.14004

[37] Kelley, John L. General topology, Graduate Texts in Mathematics, Tome 27, Springer, 1975, xiv+298 pages | MR: 370454 | Zbl: 0306.54002

[38] Kempf, George; Knudsen, Finn Faye; Mumford, David; Saint-Donat, Bernard Toroidal embeddings. I, Lecture Notes in Mathematics, Tome 339, Springer, 1973, viii+209 pages | MR: 0335518 | Zbl: 0271.14017

[39] Kleiman, Steven L. Toward a numerical theory of ampleness, Ann. Math., Tome 84 (1966), pp. 293-344 | Article | MR: 0206009 | Zbl: 0146.17001

[40] Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 48, Springer, 2004, xviii+387 pages | Article | MR: 2095471 | Zbl: 1093.14501

[41] Liu, Qing Algebraic geometry and arithmetic curves, Oxford University Press, 2006 | Zbl: 1103.14001

[42] Liu, Yifeng A non-archimedean analogue of the Calabi–Yau theorem for totally degenerate abelian varieties., J. Differ. Geom., Tome 89 (2011) no. 1, pp. 87-110 | Article | MR: 2863913 | Zbl: 1254.14026

[43] Lütkebohmert, Werner On compactification of schemes, Manuscr. Math., Tome 80 (1993) no. 1, pp. 95-111 | Article | MR: 1226600 | Zbl: 0822.14010

[44] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, Tome 8, Cambridge University Press, 1989, xiv+320 pages | MR: 1011461 | Zbl: 0666.13002

[45] Mustaţă, Mircea The non-nef locus in positive characteristic, A celebration of algebraic geometry (Clay Mathematics Proceedings) Tome 18, American Mathematical Society, 2013, pp. 535-551 | MR: 3114955 | Zbl: 1317.13011

[46] Mustaţă, Mircea; Nicaise, Johannes Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton, Algebr. Geom., Tome 2 (2015) no. 3, pp. 365-404 | Article | MR: 3370127 | Zbl: 1322.14044

[47] Pépin, Cédric Modèles semi-factoriels et modèles de Néron, Math. Ann., Tome 355 (2013) no. 1, pp. 147-185 | Article | MR: 3004579 | Zbl: 1263.14046

[48] Schwede, Karl; Tucker, Kevin A survey of test ideals, Progress in commutative algebra 2; Closures, finiteness and factorization (De Gruyter Proceedings in Mathematics), Walter de Gruyter, 2012, pp. 39-99 | MR: 2932591 | Zbl: 1254.13007

[49] The Stacks Project Authors Stacks Project, http://stacks.math.columbia.edu, 2016

[50] Thuillier, Amaury Théorie du potentiel sur les courbes en géométrie analytique non-archimédienne. Applications à la théorie d’Arakelov (2005) (Ph. D. Thesis)

[51] Yau, Shing Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, Commun. Pure Appl. Math., Tome 31 (1978), pp. 399-411 | Zbl: 0369.53059

[52] Yuan, Xinyi Big line bundles over arithmetic varieties, Invent. Math., Tome 173 (2008) no. 3, pp. 603-649 | Article | MR: 2425137 | Zbl: 1146.14016

[53] Yuan, Xinyi; Zhang, Shou-Wu The arithmetic Hodge index theorem for adelic line bundles, Math. Ann., Tome 367 (2017) no. 3-4, pp. 1123-1171 | Article | MR: 3623221 | Zbl: 1372.14017

[54] Zhang, Shou-Wu Small points and adelic metrics, J. Algebr. Geom., Tome 4 (1995) no. 2, pp. 281-300 | MR: 1311351 | Zbl: 0861.14019

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