Gubler, Walter; Martin, Florent

On Zhang's Semipositive Metrics

Doc. Math. 24, 331-372 (2019)
DOI: 10.25537/dm.2019v24.331-372

Summary

Zhang introduced semipositive metrics on a line bundle of a proper variety. In this paper, we generalize such metrics for a line bundle \(L\) of a paracompact strictly \(K\)-analytic space \(X\) over any non-archimedean field \(K\). We prove various properties in this setting such as density of piecewise \(\mathbb{Q}\)-linear metrics in the space of continuous metrics on \(L\). If \(X\) is proper scheme, then we show that algebraic, formal and piecewise linear metrics are the same. Our main result is that on a proper scheme \(X\) over an arbitrary non-archimedean field \(K\), the set of semipositive model metrics is closed with respect to pointwise convergence generalizing a result from Boucksom, Favre and Jonsson where \(K\) was assumed to be discretely valued with residue characteristic \(0\).

Mathematics Subject Classification

14G40, 14G22

Keywords/Phrases

Arakelov geometry, non-Archimedean geometry, model metrics, plurisubharmonic model functions, divisorial points

References

  • 1. Sivaramakrishna Anantharaman. Schémas en groupes, espaces homog\``enes et espaces alg\''ebriques sur une base de dimension 1. In Sur les groupes algébriques, Mém. 33 of Bull. Soc. Math. France, pages 5-79. Soc. Math. France, Paris, 1973. zbl 0286.14001; MR0335524.
  • 2. Matthew Baker, Sam Payne, and Joseph Rabinoff. Nonarchimedean geometry, tropicalization, and metrics on curves. Algebr. Geom. 3(1): 63-105 (2016). DOI 10.14231/AG-2016-004; zbl 06609386; MR3455421; arxiv 1104.0320.
  • 3. Vladimir G. Berkovich. Spectral theory and analytic geometry over non-Archimedean fields, volume 33 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1990. zbl 0715.14013; MR1070709.
  • 4. Vladimir G. Berkovich. Étale cohomology for non-Archimedean analytic spaces. Inst. Hautes Études Sci. Publ. Math., (78):5-161, 1993. DOI 10.1007/BF02712916; zbl 0804.32019; MR1259429.
  • 5. Siegfried Bosch. Lectures on formal and rigid geometry, volume 2105 of Lecture Notes in Mathematics. Springer, Cham, 2014. DOI 10.1007/978-3-319-04417-0; zbl 1314.14002; MR3309387.
  • 6. Siegfried Bosch, Ulrich Güntzer, and Reinhold Remmert. Non-Archimedean analysis. A systematic approach to rigid analytic geometry, volume 261 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1984. zbl 0539.14017; MR0746961.
  • 7. Siegfried Bosch and Werner Lütkebohmert. Formal and rigid geometry. I. Rigid spaces. Math. Ann., 295(2):291-317, 1993. DOI 10.1007/BF01444889; zbl 0808.14017; MR1202394.
  • 8. Siegfried Bosch and Werner Lütkebohmert. Formal and rigid geometry. II. Flattening techniques. Math. Ann., 296(3):403-429, 1993. DOI 10.1007/BF01445112; zbl 0808.14018; MR1225983.
  • 9. Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Formal and rigid geometry. IV. The reduced fibre theorem. Invent. Math., 119(2):361-398, 1995. DOI 10.1007/BF01245187; zbl 0839.14014; MR1312505.
  • 10. Sébastien Boucksom and Dennis Eriksson. Spaces of norms, determinant of cohomology and Fekete points in non-Archimedean geometry, 2018. arxiv 1805.01016.
  • 11. Sébastien Boucksom, Charles Favre, and Mattias Jonsson. Solution to a non-Archimedean Monge-Ampère equation. J. Amer. Math. Soc, 28(3):617-667, 2015. DOI 10.1090/S0894-0347-2014-00806-7; zbl 1325.32021; MR3327532; arxiv 1201.0188.
  • 12. Sébastien Boucksom, Charles Favre, and Mattias Jonsson. Singular semipositive metrics in non-Archimedean geometry. J. Algebraic Geom., 25(1):77-139, 2016. DOI 10.1090/jag/656; zbl 1346.14065; MR3419957; arxiv 1201.0187.
  • 13. Nicolas Bourbaki. Éléments de mathématique. Topologie générale. Chapitres 1 à 4. Hermann, Paris, 1971. zbl 1107.54001; MR0358652.
  • 14. José Ignacio Burgos Gil, Walter Gubler, Philipp Jell, Klaus Künnemann, and Florent Martin. Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld), 2016. arxiv 1608.01919.
  • 15. José Ignacio Burgos Gil, Atsushi Moriwaki, Patrice Philippon, and Martín Sombra. Arithmetic positivity on toric varieties. J. Algebraic Geom., 25(2):201-272, 2016. DOI 10.1090/jag/643; zbl 1378.14048; MR3466351; arxiv 1210.7692.
  • 16. José Ignacio Burgos Gil, Patrice Philippon, Juan Rivera-Letelier, and Martín Sombra. The distribution of Galois orbits of points of small height in toric varieties, 2015. arxiv 1509.01011.
  • 17. José Ignacio Burgos Gil, Patrice Philippon, and Martín Sombra. Arithmetic geometry of toric varieties. Metrics, measures and heights. Astérisque, (360):vi+222, 2014. zbl 1311.14050; MR3222615; arxiv 1105.5584.
  • 18. José Ignacio Burgos Gil, Patrice Philippon, and Martín Sombra. Successive minima of toric height functions. Ann. Inst. Fourier (Grenoble), 65(5):2145-2197, 2015. DOI 10.5802/aif.2985; zbl 1369.14034; MR3449209; arxiv 1403.4048.
  • 19. José Ignacio Burgos Gil, Patrice Philippon, and Martín Sombra. Height of varieties over finitely generated fields. Kyoto J. Math., 56(1):13-32, 2016. DOI 10.1215/21562261-3445138; zbl 1358.14021; MR3479316; arxiv 1408.3222.
  • 20. Antoine Chambert-Loir. Mesures et équidistribution sur les espaces de Berkovich. J. Reine Angew. Math., 595:215-235, 2006. DOI 10.1515/CRELLE.2006.049; zbl 1112.14022; MR2244803; arxiv math/0304023.
  • 21. Antoine Chambert-Loir and Antoine Ducros. Formes différentielles réelles et courants sur les espaces de Berkovich, 2012. arxiv 1204.6277.
  • 22. Antoine Chambert-Loir and Amaury Thuillier. Mesures de Mahler et équidistribution logarithmique. Ann. Inst. Fourier (Grenoble), 59(3):977-1014, 2009. DOI 10.5802/aif.2454; zbl 1192.14020; MR2543659; arxiv math/0612556.
  • 23. Brian Conrad. Deligne's notes on Nagata compactifications. J. Ramanujan Math. Soc., 22:205-257, 2007. zbl 1142.14001; MR2356346.
  • 24. Antoine Ducros. Variation de la dimension d'un morphisme analytique p-adique. Compositio Math. 143(6):1511-1532, 2007. DOI 10.1112/S0010437X07003193; zbl 1161.14018; MR2371379.
  • 25. Lawrence Ein, Tommaso de Fernex and Mircea Mustata. Vanishing theorems and singularities in birational geometry. http://homepages.math.uic.edu/~ein/DFEM.pdf.
  • 26. Xander Faber. Equidistribution of dynamically small subvarieties over the function field of a curve. Acta Arith., 137(4):345-389, 2009. DOI 10.4064/aa137-4-4; zbl 1234.37058; MR2506588; arxiv 0801.4811.
  • 27. Kazuhiro Fujiwara and Fumiharu Kato. Foundations of rigid geometry. I. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich, 2018. DOI 10.4171/135; zbl 1400.14001; MR3752648.
  • 28. Jacob Eli Goodman. Affine open subsets of algebraic varieties and ample divisors. Ann. of Math. (2), 89:160-183, 1969. DOI 10.2307/1970814; zbl 0159.50504; MR0242843.
  • 29. Alexander Grothendieck. Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes. Inst. Hautes Études Sci. Publ. Math., (8):222, 1961. DOI 10.1007/BF02684778; zbl 0118.36206; MR0217084.
  • 30. Alexander Grothendieck. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III. Inst. Hautes Études Sci. Publ. Math., (28):255, 1966. DOI 10.1007/BF02684343; zbl 0144.19904; MR0217086.
  • 31. Walter Gubler. Local heights of subvarieties over non-Archimedean fields. J. Reine Angew. Math., 498:61-113, 1998. DOI 10.1515/crll.1998.054; zbl 0906.14013; MR1629925.
  • 32. Walter Gubler. Local and canonical heights of subvarieties. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2(4):711-760, 2003. zbl 1170.14303; MR2040641.
  • 33. Walter Gubler. The Bogomolov conjecture for totally degenerate abelian varieties. Invent. Math., 169(2):377-400, 2007. DOI 10.1007/s00222-007-0049-y; zbl 1153.14029; MR2318560; arxiv math/0609387.
  • 34. Walter Gubler. Tropical varieties for non-Archimedean analytic spaces. Invent. Math., 169(2):321-376, 2007. DOI 10.1007/s00222-007-0048-z; zbl 1153.14036; MR2318559; arxiv math/0609383.
  • 35. Walter Gubler. Equidistribution over function fields. Manuscripta Math., 127(4):485-510, 2008. DOI 10.1007/s00229-008-0198-3; zbl 1189.14030; MR2457191; arxiv 0801.4508.
  • 36. Walter Gubler. A guide to tropicalizations. In Algebraic and combinatorial aspects of tropical geometry, volume 589 of Contemp. Math., pages 125-189. Amer. Math. Soc., Providence, RI, 2013. zbl 1318.14061: MR3088913; arxiv 1108.6126.
  • 37. Walter Gubler and Klaus Künnemann. A tropical approach to nonarchimedean Arakelov geometry. Algebra Number Theory, 11(1):77-180, 2017. DOI 10.2140/ant.2017.11.77; zbl 1386.14096; MR3602767; arxiv 1406.7637.
  • 38. Walter Gubler and Klaus Künnemann. Positivity properties of metrics and delta-forms. J. Reine Angew. Math. https://doi.org/10.1515/crelle-2016-0060{\tt doi.org/10.1515/crelle-2016-0060. arxiv 1509.09079; DOI 10.1515/crelle-2016-0060\tt.
  • 39. Walter Gubler, Joseph Rabinoff, and Annette Werner. Skeletons and tropicalizations. Adv. Math., 294:150-215, 2016. DOI 10.1016/j.aim.2016.02.022; zbl 1370.14024; MR3479562; arxiv 1404.7044.
  • 40. Walter Gubler, Joseph Rabinoff, and Annette Werner. Tropical Skeletons. Ann. Inst. Fourier (Grenoble), 67(5):1905-1961, 2017. DOI 10.5802/aif.3125; zbl 06984828; MR3732680; arxiv 1508.01179.
  • 41. Walter Gubler and Alejandro Soto. Classification of normal toric varieties over a valuation ring of rank one. Doc. Math., 20:171-198, 2015. https://www.elibm.org/article/10000396; zbl 1349.14161; MR3398711; arxiv 1303.1987.
  • 42. Robin Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. zbl 0367.14001; MR0463157.
  • 43. Philipp Jell. Differential forms on Berkovich analytic spaces and their cohomology. PhD thesis, Universität Regensburg, 2016. urn:nbn:de:bvb:355-epub-347884.
  • 44. Aise Johan de Jong. Smoothness, semi-stability and alterations. Inst. Hautes Études Sci. Publ. Math., (83):51-93, 1996. DOI 10.1007/BF02698644; zbl 0916.14005; MR1423020.
  • 45. Eric Katz, Joseph Rabinoff, and David Zureick-Brown. Uniform bounds for the number of rational points on curves of small Mordell-Weil rank. Duke Math. J., 165(16):3189-3240, 2016. DOI 10.1215/00127094-3673558; zbl 06666955; MR3566201; arxiv 1504.00694.
  • 46. Reinhardt Kiehl. Ein ``Descente''-Lemma und Grothendiecks Projektionssatz f\''ur nichtnoethersche Schemata. Math. Ann., 198:287-316, 1972. DOI 10.1007/BF01419561; zbl 0246.14002; MR0382280.
  • 47. Steven L. Kleiman. Toward a Numerical Theory of Ampleness. Annals of Mathematics, Second Series, Vol. 84, No. 3 (Nov. 1966), pp. 293-344. DOI 10.2307/1970447; zbl 0146.17001; MR0206009.
  • 48. Jérôme Poineau. Les espaces de Berkovich sont angéliques. Bulletin de la Société Mathématique de France 141(2):267-297, 2013. DOI 10.24033/bsmf.2648; zbl 1314.14046; MR3081557; arxiv 1105.0250.
  • 49. Michel Raynaud and Laurent Gruson. Crit\``eres de platitude et de projectivit\''e. Techniques de ``platification'' d'un module. Invent. Math., 13:1-89, 1971. DOI 10.1007/BF01390094; zbl 0227.14010; MR0308104.
  • 50. The Stacks Project Authors. Stacks Project, 2016. http://stacks.math.columbia.edu.
  • 51. Michael Temkin. On local properties of non-archimedean analytic spaces. Math. Ann., 318:585-607, 2000. DOI 10.1007/s002080000123; zbl 0972.32019; MR1800770.
  • 52. Amaury Thuillier. Théorie du potentiel sur les courbes en géométrie analytique non-archimédienne. Applications \``a la th\''eorie d'Arakelov. PhD thesis, Université de Rennes I, 2005.
  • 53. Peter Ullrich. The direct image theorem in formal and rigid geometry. Math. Ann., 301:69-104, 1995. DOI 10.1007/BF01446620; zbl 0821.32029; MR1312570.
  • 54. Paul Vojta. Nagata's embedding theorem, 2007. arxiv 0706.1907.
  • 55. Kazuhiko Yamaki. Geometric Bogomolov conjecture for abelian varieties and some results for those with some degeneration (with an appendix by Walter Gubler: the minimal dimension of a canonical measure). Manuscripta Math., 142(3-4):273-306, 2013. DOI 10.1007/s00229-012-0599-1; zbl 1281.14018; MR3117164; arxiv 1007.1081.
  • 56. Kazuhiko Yamaki. Strict supports of canonical measures and applications to the geometric Bogomolov conjecture. Compos. Math., 152(5):997-1040, 2016. DOI 10.1112/S0010437X15007721; zbl 1383.14008; MR3505646; arxiv 1211.0406.
  • 57. Xinyi Yuan. Big line bundles over arithmetic varieties. Invent. Math., 173(3):603-649, 2008. DOI 10.1007/s00222-008-0127-9; zbl 1146.14016; MR2425137; arxiv math/0612424.
  • 58. Xinyi Yuan and Shou-Wu Zhang. The arithmetic Hodge index theorem for adelic line bundles. Math. Ann., 367(3-4):1123-1171, 2017. DOI 10.1007/s00208-016-1414-1; zbl 1372.14017; MR3623221; arxiv 1304.3538.
  • 59. Shouwu Zhang. Admissible pairing on a curve. Invent. Math., 112(1):171-193, 1993. DOI 10.1007/BF01232429; zbl 0795.14015; MR1207481.
  • 60. Shouwu Zhang. Small points and adelic metrics. J. Algebraic Geom., 4(2):281-300, 1995. zbl 0861.14019; MR1311351.

Affiliation

Gubler, Walter
Fakult\``at f\''ur Mathematik, Universit\``at Regensburg, Universit\''atsstrasse 31, D-93040 Regensburg, Germany
Martin, Florent
Fakult\``at f\''ur Mathematik, Universit\``at Regensburg, Universit\''atsstrasse 31, D-93040 Regensburg, Germany

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