Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces

Mattias Jonsson; Johannes Nicaise

doi:10.5802/jep.118

Convergence of

p

-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces
[Convergence des mesures pluricanoniques p-adiques vers des mesures de Lebesgue sur des squelettes dans les espaces de Berkovich]

Mattias Jonsson ¹ ; Johannes Nicaise ²

¹ Dept of Mathematics, University of Michigan Ann Arbor, MI 48109-1043, USA
² Imperial College, Department of Mathematics South Kensington Campus, London SW72AZ, UK and KU Leuven, Department of Mathematics Celestijnenlaan 200B, 3001 Heverlee, Belgium

Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 287-336.

Résumé
Abstract

Soient $K$ un corps local non-archimédien et $X$ un $K$ -schéma lisse et propre, et fixons une forme pluricanonique sur $X$ . Pour chaque extension finie $K^{'}$ de $K$ , la forme pluricanonique induit une mesure sur la $K^{'}$ -variété analytique $X (K^{'})$ . Nous démontrons que, lorsque $K^{'}$ parcourt toutes les extensions finies modérément ramifiées de $K$ , les normalisations appropriées des images directes de ces mesures sur l’analytifié de $X$ au sens de Berkovich convergent vers une mesure de type Lebesgue sur la partie tempérée du squelette de Kontsevich-Soibelman, en supposant l’existence d’un modèle à croisements normaux stricts de $X$ . Nous démontrons également un résultat similaire pour toutes les extensions finies $K^{'}$ en supposant que $X$ admet un modèle log lisse. Il s’agit d’une version non-archimédienne de résultats analogues pour les formes de volumes sur les familles dégénérées de variétés complexes de Calabi–Yau dus à Boucksom et au premier auteur. En cours de route, nous développons une théorie générale des mesures de Lebesgue sur les squelette de Berkovich sur des corps à valuation discrète.

Let $K$ be a non-archimedean local field, $X$ a smooth and proper $K$ -scheme, and fix a pluricanonical form on $X$ . For every finite extension $K^{'}$ of $K$ , the pluricanonical form induces a measure on the $K^{'}$ -analytic manifold $X (K^{'})$ . We prove that, when $K^{'}$ runs through all finite tame extensions of $K$ , suitable normalizations of the pushforwards of these measures to the Berkovich analytification of $X$ converge to a Lebesgue-type measure on the temperate part of the Kontsevich–Soibelman skeleton, assuming the existence of a strict normal crossings model for $X$ . We also prove a similar result for all finite extensions $K^{'}$ under the assumption that $X$ has a log smooth model. This is a non-archimedean counterpart of analogous results for volume forms on degenerating complex Calabi–Yau manifolds by Boucksom and the first-named author. Along the way, we develop a general theory of Lebesgue measures on Berkovich skeleta over discretely valued fields.

Reçu le : 2019-05-15
Accepté le : 2020-01-05
Publié le : 2020-02-13

Zbl

DOI : 10.5802/jep.118

Classification : 14G22, 14J32, 32P05, 14T05
Keywords: Volume forms, local fields, Berkovich spaces
Mot clés : Formes volumes, corps locaux, espaces de Berkovich

Affiliations des auteurs :

Mattias Jonsson ¹ ; Johannes Nicaise ²

¹ Dept of Mathematics, University of Michigan Ann Arbor, MI 48109-1043, USA
² Imperial College, Department of Mathematics South Kensington Campus, London SW72AZ, UK and KU Leuven, Department of Mathematics Celestijnenlaan 200B, 3001 Heverlee, Belgium

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Mattias Jonsson and Johannes Nicaise},
     title = {Convergence of $p$-adic pluricanonical measures to {Lebesgue} measures on skeleta {in~Berkovich} spaces},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {287--336},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     doi = {10.5802/jep.118},
     zbl = {1430.14056},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.118/}
}

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Mattias Jonsson; Johannes Nicaise. Convergence of $p$-adic pluricanonical measures to Lebesgue measures on skeleta in Berkovich spaces. Journal de l’École polytechnique — Mathématiques, Tome 7 (2020), pp. 287-336. doi : 10.5802/jep.118. https://jep.centre-mersenne.org/articles/10.5802/jep.118/

Bibliographie
Cité par

[Ber90] V. G. Berkovich - Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990 | MR | Zbl

[Ber93] V. G. Berkovich - “Étale cohomology for non-Archimedean analytic spaces”, Publ. Math. Inst. Hautes Études Sci. 78 (1993), p. 5-161 (1994) | DOI | Numdam | MR | Zbl

[Ber99] V. G. Berkovich - “Smooth $p$ -adic analytic spaces are locally contractible”, Invent. Math. 137 (1999) no. 1, p. 1-84 | DOI | MR | Zbl

[BFJ16] S. Boucksom, C. Favre & M. Jonsson - “Singular semipositive metrics in non-Archimedean geometry”, J. Algebraic Geom. 25 (2016) no. 1, p. 77-139 | DOI | MR | Zbl

[BJ17] S. Boucksom & M. Jonsson - “Tropical and non-Archimedean limits of degenerating families of volume forms”, J. Éc. polytech. Math. 4 (2017), p. 87-139 | DOI | Numdam | MR | Zbl

[BLR90] S. Bosch, W. Lütkebohmert & M. Raynaud - Néron models, Ergeb. Math. Grenzgeb. (3), vol. 21, Springer-Verlag, Berlin, 1990 | DOI | Zbl

[BM19] M. V. Brown & E. Mazzon - “The essential skeleton of a product of degenerations”, Compositio Math. 155 (2019) no. 7, p. 1259-1300 | DOI | MR | Zbl

[BN16] M. Baker & J. Nicaise - “Weight functions on Berkovich curves”, Algebra Number Theory 10 (2016) no. 10, p. 2053-2079 | DOI | MR | Zbl

[BN19] E. Bultot & J. Nicaise - “Computing motivic zeta functions on log smooth models”, Math. Z. (2019), published online | DOI

[BS17] A. Bellardini & A. Smeets - “Logarithmic good reduction of abelian varieties”, Math. Ann. 369 (2017) no. 3-4, p. 1435-1442 | DOI | MR | Zbl

[Cha00] C.-L. Chai - “Néron models for semiabelian varieties: congruence and change of base field”, Asian J. Math. 4 (2000) no. 4, p. 715-736 | DOI | MR | Zbl

[CJS09] V. Cossart, U. Jannsen & S. Saito - “Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes”, 2009 | arXiv

[CLNS18] A. Chambert-Loir, J. Nicaise & J. Sebag - Motivic integration, Progress in Math., vol. 325, Birkhäuser/Springer, New York, 2018 | DOI | MR | Zbl

[CLT10] A. Chambert-Loir & Y. Tschinkel - “Igusa integrals and volume asymptotics in analytic and adelic geometry”, Confluentes Math. 2 (2010) no. 3, p. 351-429 | DOI | MR | Zbl

[CP19] V. Cossart & O. Piltant - “Resolution of singularities of arithmetical threefolds”, J. Algebra 529 (2019), p. 268-535 | DOI | MR | Zbl

[CY01] C.-L. Chai & J.-K. Yu - “Congruences of Néron models for tori and the Artin conductor (with an appendix by E. de Shalit)”, Ann. of Math. (2) 154 (2001) no. 2, p. 347-382 | DOI | Zbl

[dJ96] A. J. de Jong - “Smoothness, semi-stability and alterations”, Publ. Math. Inst. Hautes Études Sci. 83 (1996), p. 51-93 | DOI | Numdam | Zbl

[Edi92] B. Edixhoven - “Néron models and tame ramification”, Compositio Math. 81 (1992) no. 3, p. 291-306 | Numdam | MR | Zbl

[EHN15] D. Eriksson, L. H. Halle & J. Nicaise - “A logarithmic interpretation of Edixhoven’s jumps for Jacobians”, Adv. Math. 279 (2015), p. 532-574 | DOI | MR | Zbl

[HN18] L. H. Halle & J. Nicaise - “Motivic zeta functions of degenerating Calabi-Yau varieties”, Math. Ann. 370 (2018) no. 3-4, p. 1277-1320 | DOI | MR | Zbl

[Kat89] K. Kato - “Logarithmic structures of Fontaine-Illusie”, in Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, p. 191-224 | Zbl

[Kat94] K. Kato - “Toric singularities”, Amer. J. Math. 116 (1994) no. 5, p. 1073-1099 | DOI | MR | Zbl

[Kat96] F. Kato - “Log smooth deformation theory”, Tôhoku Math. J. (2) 48 (1996) no. 3, p. 317-354 | DOI | MR | Zbl

[KKMSD73] G. Kempf, F. F. Knudsen, D. Mumford & B. Saint-Donat - Toroidal embeddings. I, Lect. Notes in Math., vol. 339, Springer-Verlag, Berlin-New York, 1973 | MR | Zbl

[KM76] F. F. Knudsen & D. Mumford - “The projectivity of the moduli space of stable curves. I. Preliminaries on ‘det’ and ‘Div’”, Math. Scand. 39 (1976) no. 1, p. 19-55 | DOI | MR | Zbl

[KNX18] J. Kollár, J. Nicaise & C. Y. Xu - “Semi-stable extensions over 1-dimensional bases”, Acta Math. Sinica (N.S.) 34 (2018) no. 1, p. 103-113 | DOI | MR | Zbl

[KS04] K. Kato & T. Saito - “On the conductor formula of Bloch”, Publ. Math. Inst. Hautes Études Sci. 100 (2004), p. 5-151 | DOI | Numdam | MR | Zbl

[KS06] M. Kontsevich & Y. Soibelman - “Affine structures and non-Archimedean analytic spaces”, in The unity of mathematics, Progress in Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, p. 321-385 | DOI | MR | Zbl

[Lor10] D. Lorenzini - “Models of curves and wild ramification”, Pure Appl. Math. Q 6 (2010) no. 1, p. 41-82 | DOI | MR | Zbl

[LS03] F. Loeser & J. Sebag - “Motivic integration on smooth rigid varieties and invariants of degenerations”, Duke Math. J. 119 (2003) no. 2, p. 315-344 | DOI | MR | Zbl

[MN15] M. Mustaţă & J. Nicaise - “Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton”, Algebraic Geom. 2 (2015) no. 3, p. 365-404 | DOI | MR | Zbl

[MS17] K. Mitsui & A. Smeets - “Logarithmic good reduction and the index”, 2017 | arXiv

[Nak97] C. Nakayama - “Logarithmic étale cohomology”, Math. Ann. 308 (1997) no. 3, p. 365-404 | DOI | Zbl

[Nak98] C. Nakayama - “Nearby cycles for log smooth families”, Compositio Math. 112 (1998) no. 1, p. 45-75 | DOI | MR | Zbl

[NS07] J. Nicaise & J. Sebag - “Motivic Serre invariants, ramification, and the analytic Milnor fiber”, Invent. Math. 168 (2007) no. 1, p. 133-173 | DOI | MR | Zbl

[NX16] J. Nicaise & C. Xu - “The essential skeleton of a degeneration of algebraic varieties”, Amer. J. Math. 138 (2016) no. 6, p. 1645-1667 | DOI | MR | Zbl

[Phi94] P. Philippon - “Sur des hauteurs alternatives. II”, Ann. Inst. Fourier (Grenoble) 44 (1994) no. 4, p. 1043-1065 | DOI | Numdam | MR | Zbl

[Sai04] T. Saito - “Log smooth extension of a family of curves and semi-stable reduction”, J. Algebraic Geom. 13 (2004) no. 2, p. 287-321 | DOI | MR | Zbl

[Shi19] S. Shivaprasad - “Convergence of volume forms on a family of log-Calabi-Yau varieties to a non-Archimedean measure”, 2019 | arXiv

[Sti05] J. Stix - “A logarithmic view towards semistable reduction”, J. Algebraic Geom. 14 (2005) no. 1, p. 119-136 | DOI | MR | Zbl

[Tem16] M. Temkin - “Metrization of differential pluriforms on Berkovich analytic spaces”, in Nonarchimedean and tropical geometry, Simons Symp., Springer, 2016, p. 195-285 | DOI | MR | Zbl

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