Singular semipositive metrics in non-Archimedean geometry

Boucksom, Sébastien; Favre, Charles; Jonsson, Mattias

doi:10.1090/jag/656

Authors: Sébastien Boucksom, Charles Favre and Mattias Jonsson
Journal: J. Algebraic Geom. 25 (2016), 77-139
DOI: https://doi.org/10.1090/jag/656
Published electronically: August 14, 2015
MathSciNet review: 3419957
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Abstract | References | Additional Information

Abstract: Let $X$ be a smooth projective Berkovich space over a complete discrete valuation field $K$ of residue characteristic zero, endowed with an ample line bundle $L$. We introduce a general notion of (possibly singular) semipositive (or plurisubharmonic) metrics on $L$ and prove the analogue of the following two basic results in the complex case: the set of semipositive metrics is compact modulo scaling, and each semipositive metric is a decreasing limit of smooth semipositive ones. In particular, for continuous metrics, our definition agrees with the one by S.-W. Zhang. The proofs use multiplier ideals and the construction of suitable models of $X$ over the valuation ring of $K$, using toroidal techniques.

References

D. Abramovich, L. Caporaso, and S. Payne, The tropicalization of the moduli space of curves, to appear in Ann. Sci. Éc. Norm. Supér. arXiv:1212.0373

F. Ambro, Quasi-log varieties, Tr. Mat. Inst. Steklova 240 (2003), no. Biratsion. Geom. Lineĭn. Sist. Konechno Porozhdennye Algebry, 220–239; English transl., Proc. Steklov Inst. Math. 1(240) (2003), 214–233. MR 1993751

M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58. MR 268188

Matthew Baker and Robert Rumely, Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and Monographs, vol. 159, American Mathematical Society, Providence, RI, 2010. MR 2599526

W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984. MR 749574

Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709

Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161 (1994). MR 1259429

Vladimir G. Berkovich, Smooth $p$-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), no. 1, 1–84. MR 1702143, DOI https://doi.org/10.1007/s002220050323

Robert J. Berman, Sébastien Boucksom, Vincent Guedj, and Ahmed Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179–245. MR 3090260, DOI https://doi.org/10.1007/s10240-012-0046-6

Zbigniew Błocki and Sławomir Kołodziej, On regularization of plurisubharmonic functions on manifolds, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2089–2093. MR 2299485, DOI https://doi.org/10.1090/S0002-9939-07-08858-2

S. Bloch, H. Gillet, and C. Soulé, Non-Archimedean Arakelov theory, J. Algebraic Geom. 4 (1995), no. 3, 427–485. MR 1325788

Siegfried Bosch and Werner Lütkebohmert, Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), no. 2, 291–317. MR 1202394, DOI https://doi.org/10.1007/BF01444889

Sebastien Boucksom, Tommaso de Fernex, and Charles Favre, The volume of an isolated singularity, Duke Math. J. 161 (2012), no. 8, 1455–1520. MR 2931273, DOI https://doi.org/10.1215/00127094-1593317

S. Boucksom, T. de Fernex, C. Favre, and S. Urbinati, Valuation spaces and multiplier ideals on singular varieties, Recent Advances in Algebraic Geometry, volume in honor of Rob Lazarsfeld’s 60th birthday, London Math. Soc. Lecture Note Series, vol. 417, Cambridge Univ. Press, 2015, pp. 29–51.

Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci. 44 (2008), no. 2, 449–494. MR 2426355, DOI https://doi.org/10.2977/prims/1210167334

S. Boucksom, C. Favre, and M. Jonsson, Valuation theory in interaction, EMS Series of Congress Reports, vol. 10, European Math. Soc., Zurich, 2014, pp. 55–81.

Sébastien Boucksom, Charles Favre, and Mattias Jonsson, Solution to a non-Archimedean Monge-Ampère equation, J. Amer. Math. Soc. 28 (2015), no. 3, 617–667. MR 3327532, DOI https://doi.org/10.1090/S0894-0347-2014-00806-7

J. I. Burgos, P. Philippon, and M. Sombra, Arithmetic geometry of toric varieties, metrics, measures, and heights, Astérisque 360 (2014), 212 pages.

Antoine Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235 (French). MR 2244803, DOI https://doi.org/10.1515/CRELLE.2006.049

Antoine Chambert-Loir, Heights and measures on analytic spaces. A survey of recent results, and some remarks, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume II, London Math. Soc. Lecture Note Ser., vol. 384, Cambridge Univ. Press, Cambridge, 2011, pp. 1–50. MR 2885340

A. Chambert-Loir and A. Ducros, Formes différentielles réelles et courants sur les espaces de Berkovich, arXiv:1204.6277

Jean-Pierre Demailly, Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 87–104. MR 1178721, DOI https://doi.org/10.1007/BFb0094512

Jean-Pierre Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. MR 1158622

Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld, A subadditivity property of multiplier ideals, Michigan Math. J. 48 (2000), 137–156. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786484, DOI https://doi.org/10.1307/mmj/1030132712

Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345. MR 1257325

Antoine Ducros, Les espaces de Berkovich sont modérés (d’après Ehud Hrushovski et François Loeser), Astérisque 352 (2013), Exp. No. 1056, x, 459–507 (French, with French summary). Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043–1058. MR 3087354

A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085

Lawrence Ein, Robert Lazarsfeld, Mircea Mustaţă, Michael Nakamaye, and Mihnea Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 6, 1701–1734 (English, with English and French summaries). MR 2282673

Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith, Uniform approximation of Abhyankar valuation ideals in smooth function fields, Amer. J. Math. 125 (2003), no. 2, 409–440. MR 1963690

Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004. MR 2097722

O. Fujino, Introduction to the log minimal model program for log canonical pairs, book available at arXiv:0907.1506v1

William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037

David Gale, Victor Klee, and R. T. Rockafellar, Convex functions on convex polytopes, Proc. Amer. Math. Soc. 19 (1968), 867–873. MR 230219, DOI https://doi.org/10.1090/S0002-9939-1968-0230219-6

H. Gillet and C. Soulé, Intersection theory using Adams operations, Invent. Math. 90 (1987), no. 2, 243–277. MR 910201, DOI https://doi.org/10.1007/BF01388705

Henri Gillet and Christophe Soulé, An arithmetic Riemann-Roch theorem, Invent. Math. 110 (1992), no. 3, 473–543. MR 1189489, DOI https://doi.org/10.1007/BF01231343

Jacob Eli Goodman, Affine open subsets of algebraic varieties and ample divisors, Ann. of Math. (2) 89 (1969), 160–183. MR 242843, DOI https://doi.org/10.2307/1970814

Walter Gubler, Local heights of subvarieties over non-Archimedean fields, J. Reine Angew. Math. 498 (1998), 61–113. MR 1629925, DOI https://doi.org/10.1515/crll.1998.054

Walter Gubler, Local and canonical heights of subvarieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 4, 711–760. MR 2040641

W. Gubler, Forms and currents on the analytification of an algebraic variety (after Chambert-Loir and Ducros), to appear in Proceedings of Simons Symposia on Tropical and Non-Archimedean Geometry. arXiv:1303.7364

Vincent Guedj and Ahmed Zeriahi, Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), no. 4, 607–639. MR 2203165, DOI https://doi.org/10.1007/BF02922247

Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157

E. Hrushovski and F. Loeser, Non-Archimedean tame topology and stably dominated types, arXiv:1009.0252v3, to appear in the Annals of Mathematics Studies.

Ehud Hrushovski, François Loeser, and Bjorn Poonen, Berkovich spaces embed in Euclidean spaces, Enseign. Math. 60 (2014), no. 3-4, 273–292. MR 3342647, DOI https://doi.org/10.4171/LEM/60-3/4-4

Shuzo Izumi, A measure of integrity for local analytic algebras, Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 719–735. MR 817161, DOI https://doi.org/10.2977/prims/1195178926

Mattias Jonsson and Mircea Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2145–2209 (2013) (English, with English and French summaries). MR 3060755, DOI https://doi.org/10.5802/aif.2746

Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567–588. MR 782236, DOI https://doi.org/10.1007/BF01388524

G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518

Steven L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966), 293–344. MR 206009, DOI https://doi.org/10.2307/1970447

János Kollár, Singularities of pairs, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221–287. MR 1492525

János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959

Maxim Kontsevich and Yan Soibelman, Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 321–385. MR 2181810, DOI https://doi.org/10.1007/0-8176-4467-9_9

M. Kontsevich and Y. Tschinkel, Non-Archimedean Kähler geometry, unpublished manuscript.

Klaus Künnemann, Higher Picard varieties and the height pairing, Amer. J. Math. 118 (1996), no. 4, 781–797. MR 1400059

Aron Lagerberg, Super currents and tropical geometry, Math. Z. 270 (2012), no. 3-4, 1011–1050. MR 2892935, DOI https://doi.org/10.1007/s00209-010-0837-8

Robert Lazarsfeld, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. Positivity for vector bundles, and multiplier ideals. MR 2095472

Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232

Yifeng Liu, A non-Archimedean analogue of the Calabi-Yau theorem for totally degenerate abelian varieties, J. Differential Geom. 89 (2011), no. 1, 87–110. MR 2863913

T. Matsusaka, The criteria for algebraic equivalence and the torsion group, Amer. J. Math. 79 (1957), 53–66. MR 82730, DOI https://doi.org/10.2307/2372383

M. Mustaţǎ and J. Nicaise, Weight functions on non-archimedean analytic spaces and the Kontsevich-Soibelman skeleton, to appear in Algebraic Geometry. arXiv:1212.6328

J. Nicaise and C. Xu, The essential skeleton of a degeneration of algebraic varieties, to appear in Amer. J. Math. arXiv:1307.4041

Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894

Sam Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett. 16 (2009), no. 3, 543–556. MR 2511632, DOI https://doi.org/10.4310/MRL.2009.v16.n3.a13

Cédric Pépin, Modèles semi-factoriels et modèles de Néron, Math. Ann. 355 (2013), no. 1, 147–185 (French, with English summary). MR 3004579, DOI https://doi.org/10.1007/s00208-012-0784-2

Jérôme Poineau, Les espaces de Berkovich sont angéliques, Bull. Soc. Math. France 141 (2013), no. 2, 267–297 (French, with English and French summaries). MR 3081557, DOI https://doi.org/10.24033/bsmf.2648

Pierre Samuel, Sur les anneaux factoriels, Bull. Soc. Math. France 89 (1961), 155–173 (French). MR 139614

Michael Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Adv. Math. 219 (2008), no. 2, 488–522. MR 2435647, DOI https://doi.org/10.1016/j.aim.2008.05.006

A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne, Applications à la théorie d’Arakelov, Thesis, Université de Rennes 1, 2005, available at tel.archives-ouvertes.fr/docs/00/04/87/50/PDF/tel-00010990.pdf

Amaury Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscripta Math. 123 (2007), no. 4, 381–451 (French, with English summary). MR 2320738, DOI https://doi.org/10.1007/s00229-007-0094-2

A. Thuillier, Toroidal deformations and the homotopy type of Berkovich spaces, Seminar at the Institute of Mathematics of Bordeaux, 2011.

Michel Vaquié, Valuations, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 539–590 (French). MR 1748635

Xinyi Yuan, Big line bundles over arithmetic varieties, Invent. Math. 173 (2008), no. 3, 603–649. MR 2425137, DOI https://doi.org/10.1007/s00222-008-0127-9

X. Yuan and S.-W. Zhang, Calabi-Yau theorem and algebraic dynamics, preprint, 2009, available at www.math.columbia.edu/$\sim$szhang/papers/Preprints.htm

X. Yuan and S.-W. Zhang, The arithmetic Hodge Theorem for adelic line bundles I, arXiv:1304.3538

X. Yuan and S.-W. Zhang, The arithmetic Hodge Theorem for adelic line bundles II, arXiv:1304.3539

Shouwu Zhang, Small points and adelic metrics, J. Algebraic Geom. 4 (1995), no. 2, 281–300. MR 1311351

Additional Information

Sébastien Boucksom
Affiliation: CNRS-Université Paris 6, Institut de Mathématiques, F-75251 Paris Cedex 05, France
Address at time of publication: CNRS-CMLS, École Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID: 688226
Email: sebastien.boucksom@polytechnique.edu

Charles Favre
Affiliation: CNRS-CMLS, École Polytechnique, 91128 Palaiseau Cedex, France
MR Author ID: 641179
Email: charles.favre@polytechnique.edu

Mattias Jonsson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
MR Author ID: 631360
Email: mattiasj@umich.edu

Received by editor(s): November 9, 2012
Received by editor(s) in revised form: September 24, 2013, October 23, 2013, and December 3, 2013
Published electronically: August 14, 2015
Additional Notes: The first author was partially supported by the ANR grants MACK and POSITIVE. The second author was partially supported by the ANR-grant BERKO and the ERC-Starting grant “Nonarcomp” No. 307856. The third author was partially supported by the CNRS and the NSF. The authors’ work was carried out at several institutions, including the IHES, IMJ, the École Polytechnique, and the University of Michigan. The authors gratefully acknowledge their support.
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