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.
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@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{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {117--147},
publisher = {\'Ecole polytechnique},
volume = {5},
year = {2018},
doi = {10.5802/jep.65},
zbl = {06988575},
mrnumber = {3732694},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.65/}
}
TY - JOUR AU - Stéphane Druel AU - Henri Guenancia TI - A decomposition theorem for smoothable varieties with trivial canonical class JO - Journal de l’École polytechnique — Mathématiques PY - 2018 SP - 117 EP - 147 VL - 5 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.65/ DO - 10.5802/jep.65 LA - en ID - JEP_2018__5__117_0 ER -
%0 Journal Article %A Stéphane Druel %A Henri Guenancia %T A decomposition theorem for smoothable varieties with trivial canonical class %J Journal de l’École polytechnique — Mathématiques %D 2018 %P 117-147 %V 5 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.65/ %R 10.5802/jep.65 %G en %F JEP_2018__5__117_0
Stéphane Druel; Henri Guenancia. 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/articles/10.5802/jep.65/
[Arm82] - “Calculating the fundamental group of an orbit space”, Proc. Amer. Math. Soc. 84 (1982) no. 2, p. 267-271 | DOI | MR | Zbl
[Art76] - Lectures on deformations of singularities, Lectures on Mathematics and Physics, vol. 54, Tata Institute of Fundamental Research, Bombay, 1976
[BCHM10] - “Existence of minimal models for varieties of log general type”, J. Amer. Math. Soc. 23 (2010) no. 2, p. 405-468 | DOI | MR | Zbl
[Bea83] - “Variétés kählériennes dont la première classe de Chern est nulle”, J. Differential Geom. 18 (1983) no. 4, p. 755-782 | DOI | Zbl
[Bes87] - Einstein manifolds, Ergeb. Math. Grenzgeb. (3), vol. 10, Springer-Verlag, Berlin, 1987 | MR | Zbl
[BL04] - Complex abelian varieties, Springer, Berlin, 2004 | DOI | Zbl
[BLR90] - Néron models, Ergeb. Math. Grenzgeb. (3), vol. 21, Springer-Verlag, Berlin, 1990 | Zbl
[Bou61] - 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
[Bri10] - “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 | Zbl
[BS76] - Algebraic methods in the global theory of complex spaces, Editura Academiei; John Wiley & Sons, Bucharest; London-New York-Sydney, 1976 | Zbl
[Con00] - Grothendieck duality and base change, Lect. Notes in Math., vol. 1750, Springer-Verlag, Berlin, 2000 | MR | Zbl
[DG67] - “Critéres différentiels de régularité pour les localisés des algèbres analytiques”, J. Algebra 5 (1967), p. 305-324 | Zbl
[DG11] - 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 | DOI | Zbl
[Dru17] - “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
[DS14] - “Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry”, Acta Math. 213 (2014) no. 1, p. 63-106 | Zbl
[EGZ09] - “Singular Kähler-Einstein metrics”, J. Amer. Math. Soc. 22 (2009) no. 3, p. 607-639 | DOI | Zbl
[GGK17] - “Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups” (2017), arXiv:1704.01408
[GKKP11] - “Differential forms on log canonical spaces”, Publ. Math. Inst. Hautes Études Sci. 114 (2011), p. 87-169 | DOI | MR
[GKP16] - “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 | DOI | MR | Zbl
[Gro61] - “Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I”, Publ. Math. Inst. Hautes Études Sci. 11 (1961)
[Gro65] - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II”, Publ. Math. Inst. Hautes Études Sci. 24 (1965) | Zbl
[Gro66] - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III”, Publ. Math. Inst. Hautes Études Sci. 28 (1966) | Zbl
[Gro67] - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV”, Publ. Math. Inst. Hautes Études Sci. 32 (1967) | Zbl
[Gro95a] - “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
[Gro95b] - “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
[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
[Gro05] - 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
[Har77] - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, New York, 1977 | Zbl
[Har80] - “Stable reflexive sheaves”, Math. Ann. 254 (1980) no. 2, p. 121-176 | MR | Zbl
[Kal01] - “Symplectic resolutions: deformations and birational maps” (2001), arXiv:0012008
[Kar00] - “Minimal models and boundedness of stable varieties”, J. Algebraic Geom. 9 (2000) no. 1, p. 93-109 | MR | Zbl
[Kaw85] - “Minimal models and the Kodaira dimension of algebraic fiber spaces”, J. reine angew. Math. 363 (1985), p. 1-46 | MR | Zbl
[KKMSD73] - Toroidal embeddings. I, Lect. Notes in Math., vol. 339, Springer-Verlag, Berlin-New York, 1973 | MR | Zbl
[KM92] - “Classification of three-dimensional flips”, J. Amer. Math. Soc. 5 (1992) no. 3, p. 533-703 | DOI | MR | Zbl
[KM98] - Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998 | MR
[KMM87] - “Introduction to the minimal model problem”, in Algebraic geometry (Sendai, 1985), Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, p. 283-360 | DOI | MR | Zbl
[Kol86] - “Higher direct images of dualizing sheaves. II”, Ann. of Math. (2) 124 (1986) no. 1, p. 171-202 | MR | Zbl
[Kol93] - “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math. 113 (1993) no. 1, p. 177-215 | MR | Zbl
[Kol97] - “Singularities of pairs”, in Algebraic geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., vol. 62, American Mathematical Society, Providence, RI, 1997, p. 221-287 | DOI | MR | Zbl
[Laz04] - Positivity in algebraic geometry. I, Ergeb. Math. Grenzgeb. (3), vol. 48, Springer-Verlag, Berlin, 2004 | MR
[LWX14] - “On proper moduli spaces of smoothable Kähler-Einstein Fano varieties” (2014), arXiv:1411.0761
[MFK94] - Geometric invariant theory, Ergeb. Math. Grenzgeb. (2), vol. 34, Springer-Verlag, Berlin, 1994 | MR | Zbl
[Nak04] - Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004 | MR | Zbl
[Nam94] - “On deformations of Calabi-Yau 3-folds with terminal singularities”, Topology 33 (1994) no. 3, p. 429-446 | DOI | MR | Zbl
[Nam01] - “Deformation theory of singular symplectic -folds”, Math. Ann. 319 (2001) no. 3, p. 597-623 | MR
[Nam06] - “On deformations of -factorial symplectic varieties”, J. reine angew. Math. 599 (2006), p. 97-110 | MR | Zbl
[NS95] - “Global smoothing of Calabi-Yau threefolds”, Invent. Math. 122 (1995) no. 2, p. 403-419 | MR | Zbl
[RZ11a] - “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 | DOI | MR | Zbl
[RZ11b] - “Convergence of Calabi-Yau manifolds”, Adv. in Math. 228 (2011) no. 3, p. 1543-1589 | DOI | MR | Zbl
[Sch71] - “Rigidity of quotient singularities”, Invent. Math. 14 (1971), p. 17-26 | DOI | MR | Zbl
[Sch88] - “On fiber products of rational elliptic surfaces with section”, Math. Z. 197 (1988) no. 2, p. 177-199 | MR | Zbl
[Ser01] - Exposés de séminaires (1950-1999), Documents mathématiques, vol. 1, Société Mathématique de France, Paris, 2001 | Zbl
[SSY16] - “Existence and deformations of Kähler–Einstein metrics on smoothable -Fano varieties”, Duke Math. J. 165 (2016) no. 16, p. 3043-3083 | DOI | Zbl
[Tak03] - “Local simple connectedness of resolutions of log-terminal singularities”, Internat. J. Math. 14 (2003) no. 8, p. 825-836 | DOI | MR | Zbl
[Yau78] - “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 | DOI | Zbl
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