The Miyaoka-Yau inequality and uniformisation of canonical models

Greb, Daniel; Kebekus, Stefan; Peternell, Thomas; Taji, Behrouz

doi:10.24033/asens.2414

The Miyaoka-Yau inequality and uniformisation of canonical models
[L'inégalité de Miyaoka-Yau et l'uniformisation des modèles canoniques]

Greb, Daniel ; Kebekus, Stefan ; Peternell, Thomas ; Taji, Behrouz

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1487-1535.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé
Abstract

Nous établissons l'inégalité de Miyaoka-Yau en termes de classes de Chern orbifoldes pour le faisceau tangent d'une variété complexe projective de type général à singularités klt et diviseur canonique nef. Dans le cas d'égalité pour une variété à singularités terminales, nous établissons que le modèle canonique associé est un quotient de la boule unité par un groupe agissant discrètement.

We establish the Miyaoka-Yau inequality in terms of orbifold Chern classes for the tangent sheaf of any complex projective variety of general type with klt singularities and nef canonical divisor. In case equality is attained for a variety with at worst terminal singularities, we prove that the associated canonical model is the quotient of the unit ball by a discrete group action.

MR Zbl

DOI : 10.24033/asens.2414

Classification : 32Q30, 14E05, 32Q26, 14E05, 14E20, 14E30, 53B10, 53C07, 14C15, 14C17, 14M05.
Keywords: Classification Theory, Uniformization, Ball Quotients, Minimal Models of General Type, Miyaoka-Yau inequality, Higgs Sheaves, KLT Singularities, Canonical Models, Stability, Hyperbolicity, Flat Vector Bundles.
Mot clés : Théorie de la classification, uniformisation, quotients de la boule, modèles minimaux de type général, inégalité de Miyaoka-Yau, singularités klt, modèles canoniques, stabilité, hyperbolicité, fibrés vectoriels plats.

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     author = {Greb, Daniel and Kebekus, Stefan and Peternell, Thomas and Taji, Behrouz},
     title = {The {Miyaoka-Yau} inequality and uniformisation of canonical models},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1487--1535},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 52},
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     mrnumber = {4061021},
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     url = {http://www.numdam.org/articles/10.24033/asens.2414/}
}

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Greb, Daniel; Kebekus, Stefan; Peternell, Thomas; Taji, Behrouz. The Miyaoka-Yau inequality and uniformisation of canonical models. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 6, pp. 1487-1535. doi : 10.24033/asens.2414. http://www.numdam.org/articles/10.24033/asens.2414/

Bibliographie
Cité par

Alperin, R. C. An elementary account of Selberg's lemma, Enseign. Math., Volume 33 (1987), pp. 269-273 (ISSN: 0013-8584) | MR | Zbl

Baily, W. L. On the quotient of an analytic manifold by a group of analytic homeomorphisms, Proc. Nat. Acad. Sci. U.S.A., Volume 40 (1954), pp. 804-808 (ISSN: 0027-8424) | DOI | MR | Zbl

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), pp. 405-468 (ISSN: 0894-0347) | DOI | MR | Zbl

Beltrametti, M. C.; Sommese, A. J., De Gruyter Expositions in Mathematics, 16 | MR | Zbl

Carlson, J.; Müller-Stach, S.; Peters, C., Cambridge Studies in Advanced Math., 85, Cambridge Univ. Press, 2003, 430 pages (ISBN: 0-521-81466-9) | MR | Zbl

Calabi, E.; Vesentini, E. On compact, locally symmetric Kähler manifolds, Ann. of Math., Volume 71 (1960), pp. 472-507 (ISSN: 0003-486X) | DOI | MR | Zbl

Cheng, S. Y.; Yau, S. T., Complex differential geometry and nonlinear differential equations (Brunswick, Maine, 1984) (Contemp. Math.), Volume 49, Amer. Math. Soc., 1986, pp. 31-44 | DOI | MR | Zbl

Eyssidieux, P.; Guedj, V.; Zeriahi, A. Singular Kähler-Einstein metrics, J. Amer. Math. Soc., Volume 22 (2009), pp. 607-639 (ISSN: 0894-0347) | DOI | MR | Zbl

Eisenbud, D., Graduate Texts in Math., 150, Springer, 1995, 785 pages (ISBN: 0-387-94268-8; 0-387-94269-6) | DOI | MR | Zbl

Enoki, I., Geometry and analysis on manifolds (Katata/Kyoto, 1987) (Lecture Notes in Math.), Volume 1339, Springer, 1988, pp. 118-126 | DOI | MR | Zbl

Fulton, W.; Lazarsfeld, R., Algebraic geometry (Chicago, Ill., 1980) (Lecture Notes in Math.), Volume 862, Springer, 1981, pp. 26-92 | DOI | MR | Zbl

Flenner, H. Restrictions of semistable bundles on projective varieties, Comment. Math. Helv., Volume 59 (1984), pp. 635-650 (ISSN: 0010-2571) | DOI | MR | Zbl

Fulton, W., Ergebn. Math. Grenzg., 2, Springer, 1998, 470 pages (ISBN: 3-540-62046-X; 0-387-98549-2) | DOI | MR | Zbl

Gillet, H. Intersection theory on algebraic stacks and $Q$ -varieties, J. Pure Appl. Algebra (Proceedings of the Luminy conference on algebraic K $K$ -theory (Luminy, 1983)), Volume 34 (1984), pp. 193-240 (ISSN: 0022-4049) | DOI | MR | Zbl

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 (an extended version with additional graphics is available as arXiv:1003.2913 ) (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Greb, D.; Kebekus, S.; Peternell, T. Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J., Volume 165 (2016), pp. 1965-2004 (ISSN: 0012-7094) | DOI | MR | Zbl

Greb, D.; Kebekus, S.; Peternell, T. Movable curves and semistable sheaves, Int. Math. Res. Not., Volume 2016 (2016), pp. 536-570 (ISSN: 1073-7928) | DOI | MR | Zbl

Greb, D.; Kebekus, S.; Peternell, T.; Taji, B. Harmonic metrics on Higgs sheaves and uniformization of varieties of general type, Math. Annalen (2019) (ISSN: 1432-1807) | DOI | MR | Zbl

Goresky, M.; MacPherson, R., Ergebn. Math. Grenzg., 14, Springer, 1988, 272 pages (ISBN: 3-540-17300-5) | DOI | MR | Zbl

Greb, D.; Rollenske, S. Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities, Math. Res. Lett., Volume 18 (2011), pp. 1259-1269 (ISSN: 1073-2780) | DOI | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math., Volume 4 (1960), pp. 5-228 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math., Volume 28 (1966), pp. 5-255 (ISSN: 0073-8301) | Numdam | MR | Zbl

Grothendieck, A. Représentations linéaires et compactification profinie des groupes discrets, Manuscripta Math., Volume 2 (1970), pp. 375-396 (ISSN: 0025-2611) | DOI | MR | Zbl

Guenancia, H.; Taji, B. Orbifold stability and Miyaoka-Yau inequality for minimal pairs (preprint arXiv:1611.05981 ) | MR

Guenancia, H. Semistability of the tangent sheaf of singular varieties, Algebr. Geom., Volume 3 (2016), pp. 508-542 | DOI | MR | Zbl

Hartshorne, R., Graduate Texts in Math., 52, Springer, 1977, 496 pages (ISBN: 0-387-90244-9) | MR | Zbl

Hartshorne, R. Stable reflexive sheaves, Math. Ann., Volume 254 (1980), pp. 121-176 (ISSN: 0025-5831) | DOI | MR | Zbl

Huybrechts, D.; Lehn, M., Cambridge Mathematical Library, Cambridge Univ. Press, 2010, 325 pages (ISBN: 978-0-521-13420-0) | DOI | MR | Zbl

Holmann, H. Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann., Volume 150 (1963), pp. 327-360 (ISSN: 0025-5831) | DOI | MR | Zbl

Katok, S., Chicago Lectures in Mathematics, University of Chicago Press, 1992, 175 pages (ISBN: 0-226-42582-7; 0-226-42583-5) | MR | Zbl

Kawamata, Y. Abundance theorem for minimal threefolds, Invent. math., Volume 108 (1992), pp. 229-246 (ISSN: 0020-9910) | DOI | MR | Zbl

Kebekus, S. Pull-back morphisms for reflexive differential forms, Adv. Math., Volume 245 (2013), pp. 78-112 (ISSN: 0001-8708) | DOI | MR | Zbl

Keum, J. A fake projective plane with an order 7 automorphism, Topology, Volume 45 (2006), pp. 919-927 (ISSN: 0040-9383) | DOI | MR | Zbl

Keum, J. Quotients of fake projective planes, Geom. Topol., Volume 12 (2008), pp. 2497-2515 (ISSN: 1465-3060) | DOI | MR | Zbl

Kollár, J.; Mori, S., Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, 1998, 254 pages (ISBN: 0-521-63277-3) | DOI | MR | Zbl

Kobayashi, R.; Nakamura, S.; Sakai, F. A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A Math. Sci., Volume 65 (1989), pp. 238-241 http://projecteuclid.org/euclid.pja/1195512773 (ISSN: 0386-2194) | DOI | MR | Zbl

Kobayashi, S., World Scientific Publishing Co. Pte. Ltd., 2005, 148 pages (ISBN: 981-256-496-9) | DOI | MR

Kobayashi, R. Einstein-Kähler $V$ -metrics on open Satake $V$ -surfaces with isolated quotient singularities, Math. Ann., Volume 272 (1985), pp. 385-398 (ISSN: 0025-5831) | DOI | MR | Zbl

Kobayashi, S., Grundl. math. Wiss., 318, Springer, 1998, 471 pages (ISBN: 3-540-63534-3) | DOI | MR | Zbl

Kollár, J., Astérisque, 211, 1992

Kollár, J., M. B. Porter Lectures, Princeton Univ. Press, 1995, 201 pages (ISBN: 0-691-04381-7) | DOI | MR | Zbl

Langer, A., Algebraic geometry, de Gruyter, 2002, pp. 237-256 | DOI | MR | Zbl

Langer, A. Semistable sheaves in positive characteristic, Ann. of Math., Volume 159 (2004), pp. 251-276 (ISSN: 0003-486X) | DOI | MR | Zbl

Langer, A. Bogomolov's inequality for Higgs sheaves in positive characteristic, Invent. math., Volume 199 (2015), pp. 889-920 (ISSN: 0020-9910) | DOI | MR | Zbl

Lee, J. M., Graduate Texts in Math., 202, Springer, 2011, 433 pages (ISBN: 978-1-4419-7939-1) | DOI | MR | Zbl

Lu, S.; Taji, B. A characterization of finite quotients of abelian varieties, Int. Math. Res. Not., Volume 2018 (2018), pp. 292-319 (ISSN: 1073-7928) | DOI | MR | Zbl

Mumford, D.; Fogarty, J.; Kirwan, F., Ergebn. Math. Grenzg., 34, Springer, 1994, 292 pages (ISBN: 3-540-56963-4) | DOI | MR | Zbl

Milne, J. S., Princeton Mathematical Series, 33, Princeton Univ. Press, N.J., 1980, 323 pages (ISBN: 0-691-08238-3) | MR | Zbl

Mochizuki, T. Astérisque, Astérisque, 309, 2006, 117 pages (ISBN: 978-2-85629-226-6, ISSN: 0303-1179) | Numdam | MR | Zbl

Mumford, D., Arithmetic and geometry, Vol. II (Progr. Math.), Volume 36, Birkhäuser, 1983, pp. 271-328 | DOI | MR | Zbl

Nagata, M. On the purity of branch loci in regular local rings, Illinois J. Math., Volume 3 (1959), pp. 328-333 http://projecteuclid.org/euclid.ijm/1255455255 (ISSN: 0019-2082) | DOI | MR | Zbl

Nemirovski, S. Levi problem and semistable quotients, Complex Var. Elliptic Equ., Volume 58 (2013), pp. 1517-1525 (ISSN: 1747-6933) | DOI | MR | Zbl

Shepherd-Barron, N. I.; Wilson, P. M. H. Singular threefolds with numerically trivial first and second Chern classes, J. Algebraic Geom., Volume 3 (1994), pp. 265-281 (ISSN: 1056-3911) | MR | Zbl

Shimura, G. Algebraic varieties without deformation and the Chow variety, J. Math. Soc. Japan, Volume 20 (1968), pp. 336-341 (ISSN: 0025-5645) | DOI | MR | Zbl

Simpson, C. Local systems on proper algebraic $V$ -manifolds, Pure Appl. Math. Q., Volume 7 (2011), pp. 1675-1759 (ISSN: 1558-8599) | DOI | MR | Zbl

Simpson, C. T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., Volume 1 (1988), pp. 867-918 (ISSN: 0894-0347) | DOI | MR | Zbl

Simpson, C. T. Nonabelian Hodge theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo (1991), pp. 747-756 | MR | Zbl

Simpson, C. T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., Volume 75 (1992), pp. 5-95 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Sugiyama, K., Recent topics in differential and analytic geometry (Adv. Stud. Pure Math.), Volume 18, Academic Press, 1990, pp. 417-433 | DOI | MR | Zbl

Takayama, S. Local simple connectedness of resolutions of log-terminal singularities, Internat. J. Math., Volume 14 (2003), pp. 825-836 (ISSN: 0129-167X) | DOI | MR | Zbl

Tsuji, H. Stability of tangent bundles of minimal algebraic varieties, Topology, Volume 27 (1988), pp. 429-442 (ISSN: 0040-9383) | DOI | MR | Zbl

Tian, G.; Zhang, Z. On the Kähler-Ricci flow on projective manifolds of general type, Chinese Ann. Math. Ser. B, Volume 27 (2006), pp. 179-192 (ISSN: 0252-9599) | DOI | MR | Zbl

Vinberg, E. B.; Gorbatsevich, V. V.; Shvartsman, O. V., Lie groups and Lie algebras, II (Encyclopaedia Math. Sci.), Volume 21, Springer, 2000 | MR

Viehweg, E., Ergebn. Math. Grenzg., 30, Springer, 1995, 320 pages (ISBN: 3-540-59255-5) | DOI | MR | Zbl

Voisin, C., Cambridge Studies in Advanced Math., 76, Cambridge Univ. Press, 2007, 322 pages (ISBN: 978-0-521-71801-1) | MR | Zbl

Yau, S. T. Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A., Volume 74 (1977), pp. 1798-1799 (ISSN: 0027-8424) | DOI | MR | Zbl

Zariski, O. On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A., Volume 44 (1958), pp. 791-796 (ISSN: 0027-8424) | DOI | MR | Zbl

Zhang, Z. On degenerate Monge-Ampère equations over closed Kähler manifolds, Int. Math. Res. Not., Volume 2006 (2006) (ISSN: 1073-7928) | DOI | MR | Zbl

Zhang, Y. Miyaoka-Yau inequality for minimal projective manifolds of general type, Proc. Amer. Math. Soc., Volume 137 (2009), pp. 2749-2754 (ISSN: 0002-9939) | DOI | MR | Zbl

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