A decomposition theorem for smoothable varieties with trivial canonical class
[Un théorème de décomposition pour les variétés à singularités lissables dont la première classe de Chern est nulle]
Journal de l’École polytechnique — Mathématiques, Tome 5 (2018) , pp. 117-147.

Nous montrons que toute variété complexe projective, à singularités klt lissables et lisse en codimension deux, dont le diviseur canonique est numériquement trivial, admet un revêtement quasi-étale fini qui se décompose en un produit d’une variété abélienne et d’analogues singuliers des variétés symplectiques irréductibles et des variétés de Calabi-Yau irréductibles.

In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.

Reçu le : 2017-05-10
Accepté le : 2017-11-17
Publié le : 2017-11-29
DOI : https://doi.org/10.5802/jep.65
Classification : 14J32,  14E30
Mots clés: Variétés dont le diviseur canonique est trivial, singularités klt lissables, métriques de Kähler-Einstein sur les espaces lissables
@article{JEP_2018__5__117_0,
     author = {St\'ephane Druel and Henri Guenancia},
     title = {A decomposition theorem for smoothable~varieties with trivial~canonical~class},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     pages = {117--147},
     publisher = {\'Ecole polytechnique},
     volume = {5},
     year = {2018},
     doi = {10.5802/jep.65},
     zbl = {06988575},
     mrnumber = {3732694},
     language = {en},
     url = {jep.centre-mersenne.org/item/JEP_2018__5__117_0/}
}
Druel, Stéphane; Guenancia, Henri. A decomposition theorem for smoothable varieties with trivial canonical class. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018) , pp. 117-147. doi : 10.5802/jep.65. https://jep.centre-mersenne.org/item/JEP_2018__5__117_0/

[Arm82] M. A. Armstrong - “Calculating the fundamental group of an orbit space”, Proc. Amer. Math. Soc. 84 (1982) no. 2, p. 267-271 | Article | MR 637181 | Zbl 0505.57017

[Art76] M. Artin - Lectures on deformations of singularities, Lectures on Mathematics and Physics, vol. 54, Tata Institute of Fundamental Research, Bombay, 1976

[BCHM10] C. Birkar, P. Cascini, C. D. Hacon & J. McKernan - “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc. 23 (2010) no. 2, p. 405-468 | Article | MR 2601039 | Zbl 1210.14019

[Bea83] A. Beauville - “Variétés kählériennes dont la première classe de Chern est nulle”, J. Differential Geom. 18 (1983) no. 4, p. 755-782 | Article | Zbl 0537.53056

[Bes87] A. L. Besse - Einstein manifolds, Ergeb. Math. Grenzgeb. (3), vol. 10, Springer-Verlag, Berlin, 1987 | MR 867684 | Zbl 0613.53001

[BL04] C. Birkenhake & H. Lange - Complex abelian varieties, Springer, Berlin, 2004 | Article | Zbl 1056.14063

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

[Bou61] N. Bourbaki - Éléments de mathématique. Fascicule XXVIII. Algèbre commutative. Chapitre 3: Graduations, filtrations et topologies. Chapitre 4: Idéaux premiers associés et décomposition primaire, Actualités Scientifiques et Industrielles, vol. 1293, Hermann, Paris, 1961 | Zbl 0119.03603

[Bri10] M. Brion - “Some basic results on actions of nonaffine algebraic groups”, in Symmetry and spaces, Progress in Math., vol. 278, Birkhäuser Boston, Inc., Boston, MA, 2010, p. 1-20 | MR 2562620 | Zbl 1217.14029

[BS76] C. Bănică & O. Stănăşilă - Algebraic methods in the global theory of complex spaces, Editura Academiei; John Wiley & Sons, Bucharest; London-New York-Sydney, 1976

[Con00] B. Conrad - Grothendieck duality and base change, Lect. Notes in Math., vol. 1750, Springer-Verlag, Berlin, 2000 | MR 1804902 | Zbl 0992.14001

[DG67] J. Dieudonné & A. Grothendieck - “Critéres différentiels de régularité pour les localisés des algèbres analytiques”, J. Algebra 5 (1967), p. 305-324 | Zbl 1241.14002

[DG11] M. Demazure & A. Grothendieck - Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, Documents mathématiques, vol. 7, Société Mathématique de France, Paris, 2011, Revised and annotated edition of the 1970 original | Article | Zbl 0163.03201

[Dru17] S. Druel - “A decomposition theorem for singular spaces with trivial canonical class of dimension at most five”, Invent. Math. (2017), doi:10.1007/s00222-017-0748-y | MR 3742759

[DS14] S. Donaldson & S. Sun - “Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry”, Acta Math. 213 (2014) no. 1, p. 63-106 | Zbl 1318.53037

[EGZ09] P. Eyssidieux, V. Guedj & A. Zeriahi - “Singular Kähler-Einstein metrics”, J. Amer. Math. Soc. 22 (2009) no. 3, p. 607-639 | Article | Zbl 1215.32017

[GGK17] D. Greb, H. Guenancia & S. Kebekus - “Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups” (2017), arXiv:1704.01408

[GKKP11] D. Greb, S. Kebekus, S. J. Kovács & T. Peternell - “Differential forms on log canonical spaces”, Publ. Math. Inst. Hautes Études Sci. 114 (2011), p. 87-169 | Article | MR 2854859

[GKP16] D. Greb, S. Kebekus & T. Peternell - “Singular spaces with trivial canonical class”, in Minimal models and extremal rays (Kyoto, 2011), Adv. Stud. Pure Math., vol. 70, Mathematical Society of Japan, Tokyo, 2016, p. 67-113 | Article | MR 3617779 | Zbl 1369.14052

[Gro61] A. Grothendieck - “Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I”, Publ. Math. Inst. Hautes Études Sci. 11 (1961)

[Gro65] A. Grothendieck - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II”, Publ. Math. Inst. Hautes Études Sci. 24 (1965) | Zbl 1079.14001

[Gro66] A. Grothendieck - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III”, Publ. Math. Inst. Hautes Études Sci. 28 (1966)

[Gro67] A. Grothendieck - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV”, Publ. Math. Inst. Hautes Études Sci. 32 (1967) | Zbl 0135.39701

[Gro95a] A. Grothendieck - “Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert”, in Séminaire Bourbaki (1960-61), Vol. 6, Société Mathématique de France, Paris, 1995, p. 249-276, Exp. No. 221 | Zbl 0144.19904

[Gro95b] A. Grothendieck - “Technique de descente et théorèmes d’existence en géométrie algébrique. V. Les schémas de Picard: théorèmes d’existence”, in Séminaire Bourbaki (1961-62), Vol. 7, Société Mathématique de France, Paris, 1995, p. 143-161, Exp. No. 232 | Numdam | Zbl 0153.22301

[Gro03] - Revêtements étales et groupe fondamental (SGA 1), Documents mathématiques, vol. 3, Société Mathématique de France, Paris, 2003, Updated and annotated reprint of the 1971 original | Zbl 0236.14003

[Gro05] A. Grothendieck - Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents mathématique, vol. 4, Société Mathématique de France, Paris, 2005, Revised reprint of the 1968 original | Zbl 0238.14014

[Har77] R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York, 1977 | Zbl 0367.14001

[Har80] R. Hartshorne - “Stable reflexive sheaves”, Math. Ann. 254 (1980) no. 2, p. 121-176 | MR 597077 | Zbl 0431.14004

[Kal01] D. Kaledin - “Symplectic resolutions: deformations and birational maps” (2001), arXiv:0012008

[Kar00] K. Karu - “Minimal models and boundedness of stable varieties”, J. Algebraic Geom. 9 (2000) no. 1, p. 93-109 | MR 1713521 | Zbl 0980.14008

[Kaw85] Y. Kawamata - “Minimal models and the Kodaira dimension of algebraic fiber spaces”, J. reine angew. Math. 363 (1985), p. 1-46 | MR 814013 | Zbl 0589.14014

[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

[KM92] J. Kollár & S. Mori - “Classification of three-dimensional flips”, J. Amer. Math. Soc. 5 (1992) no. 3, p. 533-703 | Article | MR 1149195 | Zbl 0773.14004

[KM98] J. Kollár & S. Mori - Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998 | MR 1658959

[KMM87] Y. Kawamata, K. Matsuda & K. Matsuki - “Introduction to the minimal model problem”, in Algebraic geometry (Sendai, 1985), Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, p. 283-360 | Article | MR 946243 | Zbl 0672.14006

[Kol86] J. Kollár - “Higher direct images of dualizing sheaves. II”, Ann. of Math. (2) 124 (1986) no. 1, p. 171-202 | MR 847955 | Zbl 0605.14014

[Kol93] J. Kollár - “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math. 113 (1993) no. 1, p. 177-215 | MR 1223229 | Zbl 0819.14006

[Kol97] J. Kollár - “Singularities of pairs”, in Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., vol. 62, American Mathematical Society, Providence, RI, 1997, p. 221-287 | Article | MR 1492525 | Zbl 0905.14002

[Laz04] R. Lazarsfeld - Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer-Verlag, Berlin, 2004 | MR 2095471

[LWX14] C. Li, X. Wang & C. Xu - “On proper moduli spaces of smoothable Kähler-Einstein Fano varieties” (2014), arXiv:1411.0761

[MFK94] D. Mumford, J. Fogarty & F. Kirwan - Geometric invariant theory, Ergeb. Math. Grenzgeb. (2), vol. 34, Springer-Verlag, Berlin, 1994 | MR 1304906 | Zbl 0797.14004

[Nak04] N. Nakayama - Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004 | MR 2104208 | Zbl 1061.14018

[Nam94] Y. Namikawa - “On deformations of Calabi-Yau 3-folds with terminal singularities”, Topology 33 (1994) no. 3, p. 429-446 | Article | MR 1819886 | Zbl 0989.53055

[Nam01] Y. Namikawa - “Deformation theory of singular symplectic n-folds”, Math. Ann. 319 (2001) no. 3, p. 597-623 | MR 2279099

[Nam06] Y. Namikawa - “On deformations of -factorial symplectic varieties”, J. reine angew. Math. 599 (2006), p. 97-110 | MR 1286924 | Zbl 0813.14004

[NS95] Y. Namikawa & J. H. M. Steenbrink - “Global smoothing of Calabi-Yau threefolds”, Invent. Math. 122 (1995) no. 2, p. 403-419 | MR 1358982 | Zbl 0861.14036

[RZ11a] X. Rong & Y. Zhang - “Continuity of extremal transitions and flops for Calabi-Yau manifolds”, J. Differential Geom. 89 (2011) no. 2, p. 233-269, Appendix B by Mark Gross | Article | MR 2863918 | Zbl 1264.32021

[RZ11b] W.-D. Ruan & Y. Zhang - “Convergence of Calabi-Yau manifolds”, Adv. in Math. 228 (2011) no. 3, p. 1543-1589 | Article | MR 2824563 | Zbl 1232.32012

[Sch71] M. Schlessinger - “Rigidity of quotient singularities”, Invent. Math. 14 (1971), p. 17-26 | Article | MR 292830 | Zbl 0232.14005

[Sch88] C. Schoen - “On fiber products of rational elliptic surfaces with section”, Math. Z. 197 (1988) no. 2, p. 177-199 | MR 923487 | Zbl 0631.14032

[Ser01] J.-P. Serre - Exposés de séminaires (1950-1999), Documents mathématiques, vol. 1, Société Mathématique de France, Paris, 2001 | Zbl 1005.01012

[SSY16] C. Spotti, S. Sun & C. Yao - “Existence and deformations of Kähler–Einstein metrics on smoothable -Fano varieties”, Duke Math. J. 165 (2016) no. 16, p. 3043-3083 | Article | Zbl 1362.53082

[Tak03] S. Takayama - “Local simple connectedness of resolutions of log-terminal singularities”, Internat. J. Math. 14 (2003) no. 8, p. 825-836 | Article | MR 2013147 | Zbl 1058.14024

[Yau78] S.-T. Yau - “On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I.”, Comm. Pure Appl. Math. 31 (1978), p. 339-411 | Article | Zbl 0369.53059