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 :
Accepté le :
Publié le :
DOI : 10.5802/jep.65
Classification : 14J32, 14E30
Keywords: Varieties with trivial canonical divisor, smoothable klt singularities, Kähler-Einstein metrics on smoothable spaces
Mot clés : Variétés dont le diviseur canonique est trivial, singularités klt lissables, métriques de Kähler-Einstein sur les espaces lissables
Druel, Stéphane 1 ; Guenancia, Henri 2

1 Institut Fourier, UMR 5582 du CNRS, Université Grenoble Alpes CS 40700, 38058 Grenoble cedex 9, France
2 Department of Mathematics, Stony Brook University Stony Brook, NY 11794-3651, United States
@article{JEP_2018__5__117_0,
     author = {Druel, St\'ephane and Guenancia, Henri},
     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},
     mrnumber = {3732694},
     zbl = {06988575},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.65/}
}
TY  - JOUR
AU  - Druel, Stéphane
AU  - Guenancia, Henri
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  - Ecole polytechnique
UR  - http://archive.numdam.org/articles/10.5802/jep.65/
DO  - 10.5802/jep.65
LA  - en
ID  - JEP_2018__5__117_0
ER  - 
%0 Journal Article
%A Druel, Stéphane
%A Guenancia, Henri
%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 Ecole polytechnique
%U http://archive.numdam.org/articles/10.5802/jep.65/
%R 10.5802/jep.65
%G en
%F 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. http://archive.numdam.org/articles/10.5802/jep.65/

[Arm82] Armstrong, M. A. Calculating the fundamental group of an orbit space, Proc. Amer. Math. Soc., Volume 84 (1982) no. 2, pp. 267-271 | DOI | MR | Zbl

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

[BCHM10] Birkar, C.; Cascini, P.; Hacon, C. D.; McKernan, J. Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR | Zbl

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

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

[BL04] Birkenhake, C.; Lange, H. Complex abelian varieties, Springer, Berlin, 2004 | DOI | Zbl

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

[Bou61] Bourbaki, N. 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, 1293, Hermann, Paris, 1961 | Zbl

[Bri10] Brion, M. Some basic results on actions of nonaffine algebraic groups, Symmetry and spaces (Progress in Math.), Volume 278, Birkhäuser Boston, Inc., Boston, MA, 2010, pp. 1-20 | MR | Zbl

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

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

[DG67] Dieudonné, J.; Grothendieck, A. Critéres différentiels de régularité pour les localisés des algèbres analytiques, J. Algebra, Volume 5 (1967), pp. 305-324 | Zbl

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

[Dru17] Druel, S. 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] Donaldson, S.; Sun, S. Gromov-Hausdorff limits of Kähler manifolds and algebraic geometry, Acta Math., Volume 213 (2014) no. 1, pp. 63-106 | Zbl

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

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

[GKKP11] Greb, D.; Kebekus, S.; Kovács, S. J.; Peternell, T. Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci., Volume 114 (2011), pp. 87-169 | DOI | MR

[GKP16] Greb, D.; Kebekus, S.; Peternell, T. Singular spaces with trivial canonical class, Minimal models and extremal rays (Kyoto, 2011) (Adv. Stud. Pure Math.), Volume 70, Mathematical Society of Japan, Tokyo, 2016, pp. 67-113 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

[Har80] Hartshorne, R. Stable reflexive sheaves, Math. Ann., Volume 254 (1980) no. 2, pp. 121-176 | MR | Zbl

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

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

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

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

[KM92] Kollár, J.; Mori, S. Classification of three-dimensional flips, J. Amer. Math. Soc., Volume 5 (1992) no. 3, pp. 533-703 | DOI | MR | Zbl

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

[KMM87] Kawamata, Y.; Matsuda, K.; Matsuki, K. Introduction to the minimal model problem, Algebraic geometry (Sendai, 1985) (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 283-360 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

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

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

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

[Nak04] Nakayama, N. Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004, xiv+277 pages | MR | Zbl

[Nam94] Namikawa, Y. On deformations of Calabi-Yau 3-folds with terminal singularities, Topology, Volume 33 (1994) no. 3, pp. 429-446 | DOI | MR | Zbl

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

[Nam06] Namikawa, Y. On deformations of -factorial symplectic varieties, J. reine angew. Math., Volume 599 (2006), pp. 97-110 | MR | Zbl

[NS95] Namikawa, Y.; Steenbrink, J. H. M. Global smoothing of Calabi-Yau threefolds, Invent. Math., Volume 122 (1995) no. 2, pp. 403-419 | MR | Zbl

[RZ11a] Rong, X.; Zhang, Y. Continuity of extremal transitions and flops for Calabi-Yau manifolds, J. Differential Geom., Volume 89 (2011) no. 2, pp. 233-269 (Appendix B by Mark Gross) | DOI | MR | Zbl

[RZ11b] Ruan, W.-D.; Zhang, Y. Convergence of Calabi-Yau manifolds, Adv. in Math., Volume 228 (2011) no. 3, pp. 1543-1589 | DOI | MR | Zbl

[Sch71] Schlessinger, M. Rigidity of quotient singularities, Invent. Math., Volume 14 (1971), pp. 17-26 | DOI | MR | Zbl

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

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

[SSY16] Spotti, C.; Sun, S.; Yao, C. Existence and deformations of Kähler–Einstein metrics on smoothable -Fano varieties, Duke Math. J., Volume 165 (2016) no. 16, pp. 3043-3083 | DOI | Zbl

[Tak03] Takayama, Shigeharu Local simple connectedness of resolutions of log-terminal singularities, Internat. J. Math., Volume 14 (2003) no. 8, pp. 825-836 | DOI | MR | Zbl

[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), pp. 339-411 | DOI | Zbl

Cité par Sources :