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

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
DOI : https://doi.org/10.5802/jep.118
Classification : 14G22,  14J32,  32P05,  14T05
Mots clés: Formes volumes, corps locaux, espaces de Berkovich
@article{JEP_2020__7__287_0,
     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'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {7},
     year = {2020},
     pages = {287-336},
     doi = {10.5802/jep.118},
     zbl = {1430.14056},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2020__7__287_0/}
}
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/item/JEP_2020__7__287_0/

[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 1070709 | Zbl 0715.14013

[Ber93] V. G. Berkovich - “Étale cohomology for non-Archimedean analytic spaces”, Publ. Math. Inst. Hautes Études Sci. 78 (1993), p. 5-161 (1994) | Article | Numdam | MR 1259429 | Zbl 0804.32019

[Ber99] V. G. Berkovich - “Smooth p-adic analytic spaces are locally contractible”, Invent. Math. 137 (1999) no. 1, p. 1-84 | Article | MR 1702143 | Zbl 0930.32016

[BFJ16] S. Boucksom, C. Favre & M. Jonsson - “Singular semipositive metrics in non-Archimedean geometry”, J. Algebraic Geom. 25 (2016) no. 1, p. 77-139 | Article | MR 3419957 | Zbl 1346.14065

[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 | Article | Numdam | MR 3611100 | Zbl 1401.32019

[BLR90] S. Bosch, W. Lütkebohmert & M. Raynaud - Néron models, Ergeb. Math. Grenzgeb. (3), vol. 21, Springer-Verlag, Berlin, 1990 | Article | Zbl 0705.14001

[BM19] M. V. Brown & E. Mazzon - “The essential skeleton of a product of degenerations”, Compositio Math. 155 (2019) no. 7, p. 1259-1300 | Article | MR 3963489 | Zbl 07067394

[BN16] M. Baker & J. Nicaise - “Weight functions on Berkovich curves”, Algebra Number Theory 10 (2016) no. 10, p. 2053-2079 | Article | MR 3582013 | Zbl 1375.14208

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

[BS17] A. Bellardini & A. Smeets - “Logarithmic good reduction of abelian varieties”, Math. Ann. 369 (2017) no. 3-4, p. 1435-1442 | Article | MR 3713546 | Zbl 1393.14017

[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 | Article | MR 1870655 | Zbl 1100.14511

[CJS09] V. Cossart, U. Jannsen & S. Saito - “Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes”, 2009 | arXiv:0905.2191

[CLNS18] A. Chambert-Loir, J. Nicaise & J. Sebag - Motivic integration, Progress in Math., vol. 325, Birkhäuser/Springer, New York, 2018 | Article | MR 3838446 | Zbl 06862764

[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 | Article | MR 2740045 | Zbl 1206.11086

[CP19] V. Cossart & O. Piltant - “Resolution of singularities of arithmetical threefolds”, J. Algebra 529 (2019), p. 268-535 | Article | MR 3942183 | Zbl 07057272

[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 | Article | Zbl 1098.14014

[dJ96] A. J. de Jong - “Smoothness, semi-stability and alterations”, Publ. Math. Inst. Hautes Études Sci. 83 (1996), p. 51-93 | Article | Numdam | Zbl 0916.14005

[Edi92] B. Edixhoven - “Néron models and tame ramification”, Compositio Math. 81 (1992) no. 3, p. 291-306 | Numdam | Zbl 0759.14033

[EHN15] D. Eriksson, L. H. Halle & J. Nicaise - “A logarithmic interpretation of Edixhoven’s jumps for Jacobians”, Adv. Math. 279 (2015), p. 532-574 | Article | MR 3345191 | Zbl 1396.11092

[HN18] L. H. Halle & J. Nicaise - “Motivic zeta functions of degenerating Calabi-Yau varieties”, Math. Ann. 370 (2018) no. 3-4, p. 1277-1320 | Article | MR 3770167 | Zbl 1400.14045

[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 0776.14004

[Kat94] K. Kato - “Toric singularities”, Amer. J. Math. 116 (1994) no. 5, p. 1073-1099 | Article | MR 1296725 | Zbl 0832.14002

[Kat96] F. Kato - “Log smooth deformation theory”, Tôhoku Math. J. (2) 48 (1996) no. 3, p. 317-354 | Article | MR 1404507 | Zbl 0876.14007

[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 335518 | Zbl 0271.14017

[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 | Article | MR 437541 | Zbl 0343.14008

[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 | Article | MR 3735836 | Zbl 1408.14061

[KS04] K. Kato & T. Saito - “On the conductor formula of Bloch”, Publ. Math. Inst. Hautes Études Sci. 100 (2004), p. 5-151 | Article | Numdam | MR 2102698 | Zbl 1099.14009

[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 | Article | MR 2181810 | Zbl 1114.14027

[Lor10] D. Lorenzini - “Models of curves and wild ramification”, Pure Appl. Math. Q 6 (2010) no. 1, p. 41-82 | Article | MR 2591187 | Zbl 1200.14052

[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 | Article | MR 1997948 | Zbl 1078.14029

[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 | Article | MR 3370127 | Zbl 1322.14044

[MS17] K. Mitsui & A. Smeets - “Logarithmic good reduction and the index”, 2017 | arXiv:1711.11547v2

[Nak97] C. Nakayama - “Logarithmic étale cohomology”, Math. Ann. 308 (1997) no. 3, p. 365-404 | Article | Zbl 0877.14016

[Nak98] C. Nakayama - “Nearby cycles for log smooth families”, Compositio Math. 112 (1998) no. 1, p. 45-75 | Article | MR 1622751 | Zbl 0926.14006

[NS07] J. Nicaise & J. Sebag - “Motivic Serre invariants, ramification, and the analytic Milnor fiber”, Invent. Math. 168 (2007) no. 1, p. 133-173 | Article | MR 2285749 | Zbl 1136.14010

[NX16] J. Nicaise & C. Xu - “The essential skeleton of a degeneration of algebraic varieties”, Amer. J. Math. 138 (2016) no. 6, p. 1645-1667 | Article | MR 3595497 | Zbl 1375.14092

[Phi94] P. Philippon - “Sur des hauteurs alternatives. II”, Ann. Inst. Fourier (Grenoble) 44 (1994) no. 4, p. 1043-1065 | Article | Numdam | MR 1306549 | Zbl 0878.11024

[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 | Article | MR 2047700 | Zbl 1082.14032

[Shi19] S. Shivaprasad - “Convergence of volume forms on a family of log-Calabi-Yau varieties to a non-Archimedean measure”, 2019 | arXiv:1911.07307

[Sti05] J. Stix - “A logarithmic view towards semistable reduction”, J. Algebraic Geom. 14 (2005) no. 1, p. 119-136 | Article | MR 2092128 | Zbl 1080.14035

[Tem16] M. Temkin - “Metrization of differential pluriforms on Berkovich analytic spaces”, in Nonarchimedean and tropical geometry, Simons Symp., Springer, 2016, p. 195-285 | Article | MR 3702313 | Zbl 1360.32019