Tropical and non-Archimedean limits of degenerating families of volume forms
[Limites tropicales et non archimédiennes de familles de formes volumes qui dégénèrent]
Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 87-139.

Nous étudions le comportement asymptotique de formes volumes dans une famille de variétés complexes compactes qui dégénèrent. Sous des conditions assez générales, nous montrons que les formes volumes convergent en un sens naturel vers une mesure du type de Lebesgue sur un certain complexe simplicial. Ceci fournit en particulier une version en théorie de la mesure d’une conjecture de Kontsevich–Soibelman et Gross–Wilson portant sur les dégénérescences maximales de variétés de Calabi-Yau.

We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich–Soibelman and Gross–Wilson, bearing on maximal degenerations of Calabi–Yau manifolds.

Reçu le :
Accepté le :
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DOI : 10.5802/jep.39
Classification : 32Q25, 14J32, 14T05, 53C23, 32P05, 14G22
Keywords: Calabi-Yau manifolds, volume forms, degenerations, Berkovich spaces
Mot clés : Variétés de Calabi-Yau, formes volumes, dégénérescences, espaces de Berkovich
Boucksom, Sébastien 1 ; Jonsson, Mattias 2

1 CMLS, École polytechnique, CNRS, Université Paris-Saclay 91128 Palaiseau Cedex, France
2 Department of Mathematics, University of Michigan Ann Arbor, MI 48109-1043, USA and Mathematical Sciences, Chalmers University of Technology and University of Gothenburg SE-412 96 Göteborg, Sweden
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Boucksom, Sébastien; Jonsson, Mattias. Tropical and non-Archimedean limits of degenerating families of volume forms. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 87-139. doi : 10.5802/jep.39. http://www.numdam.org/articles/10.5802/jep.39/

[AGZV12] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. Singularities of differentiable maps. Vol. 1, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2012 | Zbl

[BBE + 16] Berman, R. J.; Boucksom, S.; Eyssidieux, Ph.; Guedj, V.; Zeriahi, A. Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties, J. reine angew. Math. (2016) (online, arXiv:1111.7158) | DOI

[Bea83] Beauville, A. Variétés kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., Volume 18 (1983) no. 4, pp. 755-782 | DOI | Zbl

[Ber71] Bergman, G. M. The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc., Volume 157 (1971), pp. 459-469 | DOI | MR | Zbl

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

[Ber04] Berkovich, V. G. Smooth p-adic analytic spaces are locally contractible. II, Geometric aspects of Dwork theory, Walter de Gruyter GmbH & Co., Berlin, 2004, pp. 293-370 | DOI | MR | Zbl

[Ber09] Berkovich, V. G. A non-Archimedean interpretation of the weight zero subspaces of limit mixed Hodge structures, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I (Progress in Math.), Volume 269, Birkhäuser Boston, Boston, MA, 2009, pp. 49-67 | MR | Zbl

[BFJ08] Boucksom, S.; Favre, Ch.; Jonsson, M. Valuations and plurisubharmonic singularities, Publ. RIMS, Kyoto Univ., Volume 44 (2008) no. 2, pp. 449-494 | DOI | MR | Zbl

[BFJ15] Boucksom, S.; Favre, Ch.; Jonsson, M. Solution to a non-Archimedean Monge-Ampère equation, J. Amer. Math. Soc., Volume 28 (2015) no. 3, pp. 617-667 | DOI | Zbl

[BFJ16] Boucksom, S.; Favre, Ch.; Jonsson, M. Singular semipositive metrics in non-Archimedean geometry, J. Algebraic Geom., Volume 25 (2016) no. 1, pp. 77-139 | DOI | MR | Zbl

[BHJ16] Boucksom, S.; Hisamoto, T.; Jonsson, M. Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016) to appear in J. Eur. Math. Soc. (JEMS), arXiv:1603.01026

[Bog74] Bogomolov, F. A. On the decomposition of Kähler manifolds with trivial canonical class, Math. USSR-Sb., Volume 22 (1974), pp. 580-583 | DOI | Zbl

[Cle77] Clemens, C. H. Degeneration of Kähler manifolds, Duke Math. J., Volume 44 (1977) no. 2, pp. 215-290 | DOI | MR | Zbl

[CLT10] Chambert-Loir, A.; Tschinkel, Yu. Igusa integrals and volume asymptotics in analytic and adelic geometry, Confluentes Math., Volume 2 (2010) no. 3, pp. 351-429 | DOI | MR | Zbl

[dFEM11] de Fernex, T.; Ein, L.; Mustaţă, M. Log canonical thresholds on varieties with bounded singularities, Classification of algebraic varieties (EMS Ser. Congr. Rep.), European Mathematical Society, Zürich, 2011, pp. 221-257 | DOI | Zbl

[dFKX12] de Fernex, T.; Kollár, J.; Xu, C. The dual complex of singularities (2012) (to appear in Adv. Stud. Pure Math., arXiv:1212.1675)

[DS14] Donaldson, S.; Sun, S. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., Volume 213 (2014) no. 1, pp. 63-106 | DOI | Zbl

[EGZ09] Eyssidieux, Ph.; Guedj, V.; Zeriahi, A. Singular Kähler-Einstein metrics, J. Amer. Math. Soc., Volume 22 (2009) no. 3, pp. 607-639 | DOI | Zbl

[FJ04] Favre, Ch.; Jonsson, M. The valuative tree, Lect. Notes in Math., 1853, Springer-Verlag, Berlin, 2004 | MR | Zbl

[GTZ13] Gross, M.; Tosatti, V.; Zhang, Y. Collapsing of abelian fibered Calabi-Yau manifolds, Duke Math. J., Volume 162 (2013) no. 3, pp. 517-551 | DOI | MR | Zbl

[GTZ16] Gross, M.; Tosatti, V.; Zhang, Y. Gromov-Hausdorff collapsing of Calabi-Yau manifolds, Comm. Anal. Geom., Volume 24 (2016) no. 1, pp. 93-113 | DOI | MR | Zbl

[Gub98] Gubler, W. Local heights of subvarieties over non-Archimedean fields, J. reine angew. Math., Volume 498 (1998), pp. 61-113 | MR | Zbl

[GW00] Gross, M.; Wilson, P. M. H. Large complex structure limits of K3 surfaces, J. Differential Geom., Volume 55 (2000) no. 3, pp. 475-546 | DOI | MR | Zbl

[Hir75] Hironaka, H. Flattening theorem in complex-analytic geometry, Amer. J. Math., Volume 97 (1975), pp. 503-547 | DOI | MR | Zbl

[HT15] Hein, H.-J.; Tosatti, V. Remarks on the collapsing of torus fibered Calabi-Yau manifolds, Bull. London Math. Soc., Volume 47 (2015) no. 6, pp. 1021-1027 | MR | Zbl

[JM12] Jonsson, M.; Mustaţă, M. Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 6, pp. 2145-2209 | DOI | Numdam | MR | Zbl

[Jon16] Jonsson, M. Degenerations of amoebae and Berkovich spaces, Math. Ann., Volume 364 (2016) no. 1-2, pp. 293-311 | DOI | MR | Zbl

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

[KNX15] Kollár, J.; Nicaise, J.; Xu, C. Semi-stable extensions over 1-dimensional bases (2015) (arXiv:1510.02446)

[Kol97] Kollár, J. Singularities of pairs, Algebraic geometry (Santa Cruz, 1995) (Proc. Sympos. Pure Math.), Volume 62, American Mathematical Society, Providence, RI, 1997, pp. 221-287 | MR | Zbl

[Kol07] Kollár, J. Lectures on resolution of singularities, Annals of Mathematics Studies, 166, Princeton University Press, Princeton, NJ, 2007 | MR

[Kol13] Kollár, J. Singularities of the minimal model program, Cambridge Tracts in Mathematics, 200, Cambridge University Press, Cambridge, 2013 | DOI | MR | Zbl

[KS01] Kontsevich, M.; Soibelman, Y. Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000), World Sci. Publ., River Edge, NJ, 2001, pp. 203-263 | DOI | Zbl

[KS06] Kontsevich, M.; Soibelman, Y. Affine structures and non-Archimedean analytic spaces, The unity of mathematics (Progress in Math.), Volume 244, Birkhäuser Boston, Boston, MA, 2006, pp. 321-385 | DOI | MR | Zbl

[KX16] Kollár, J.; Xu, C. The dual complex of Calabi–Yau pairs, Invent. Math., Volume 205 (2016) no. 3, pp. 527-557 | DOI | MR | Zbl

[Li15] Li, C. Yau-Tian-Donaldson correspondence for K-semistable Fano manifolds, J. reine angew. Math. (2015) (online, arXiv:1302.6681) | DOI | Zbl

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

[MS84] Morgan, J. W.; Shalen, P. B. Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. (2), Volume 120 (1984) no. 3, pp. 401-476 | DOI | MR | Zbl

[NX16a] Nicaise, J.; Xu, C. Poles of maximal order of motivic zeta functions, Duke Math. J., Volume 165 (2016) no. 2, pp. 217-243 | DOI | MR | Zbl

[NX16b] Nicaise, J.; Xu, C. The essential skeleton of a degeneration of algebraic varieties, Amer. J. Math., Volume 138 (2016) no. 6, pp. 1645-1667 | DOI | MR | Zbl

[Oda14] Odaka, Y. Tropically compactify via Gromov-Hausdorff collapse (2014) (arXiv:1406.7772)

[Poi10] Poineau, J. La droite de Berkovich sur Z, Astérisque, 334, Société Mathématique de France, Paris, 2010 | MR | Zbl

[Poi13] Poineau, J. Espaces de Berkovich sur Z: étude locale, Invent. Math., Volume 194 (2013) no. 3, pp. 535-590 | DOI | MR | Zbl

[RZ11] Ruan, W.-D.; Zhang, Y. Convergence of Calabi-Yau manifolds, Advances in Math., Volume 228 (2011) no. 3, pp. 1543-1589 | DOI | MR | Zbl

[RZ13] Rong, X.; Zhang, Y. Degenerations of Ricci-flat Calabi-Yau manifolds, Commun. Contemp. Math., Volume 15 (2013) no. 4, p. 1250057, 8 | DOI | MR | Zbl

[Sch73] Schmid, W. Variation of Hodge structure: the singularities of the period mapping, Invent. Math., Volume 22 (1973), pp. 211-319 | DOI | MR | Zbl

[Tak15] Takayama, S. On moderate degenerations of polarized Ricci-flat Kähler manifolds, J. Math. Sci. Univ. Tokyo, Volume 22 (2015) no. 1, pp. 469-489 | Zbl

[Tem16] Temkin, M. Metrization of differential pluriforms on Berkovich analytic spaces, Nonarchimedean and tropical geometry (Simons Symposia), Springer International Publishing, 2016, pp. 195-285 | DOI | MR | Zbl

[Tos09] Tosatti, V. Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc. (JEMS), Volume 11 (2009) no. 4, pp. 755-776 | DOI | MR | Zbl

[Tos10] Tosatti, V. Adiabatic limits of Ricci-flat Kähler metrics, J. Differential Geom., Volume 84 (2010) no. 2, pp. 427-453 | DOI | MR | Zbl

[Tos15] Tosatti, V. Families of Calabi-Yau manifolds and canonical singularities, Internat. Math. Res. Notices (2015) no. 20, pp. 10586-10594 | DOI | MR | Zbl

[TWY14] Tosatti, V.; Weinkove, B.; Yang, X. The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014) (arXiv:1408.0161) | Zbl

[Wan03] Wang, C.-L. Quasi-Hodge metrics and canonical singularities, Math. Res. Lett., Volume 10 (2003) no. 1, pp. 57-70 | DOI | MR | Zbl

[Wło09] Włodarczyk, J. Resolution of singularities of analytic spaces, Proceedings of Gökova Geometry-Topology Conference 2008, Gökova Geometry/Topology Conference (GGT), Gökova (2009), pp. 31-63

[Yau78] Yau, S. T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | DOI | Zbl

[ZS75] Zariski, O.; Samuel, P. Commutative algebra. Vol. II, Graduate Texts in Math., 29, Springer-Verlag, 1975 | MR

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