The Miyaoka-Yau inequality and uniformisation of canonical models

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

doi:10.24033/asens.2414

Geodesic

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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.

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Abstract (VO)
Résumé

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.

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.

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.
Mots-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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     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},
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     mrnumber = {4061021},
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     url = {http://geodesic.mathdoc.fr/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://geodesic.mathdoc.fr/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) | MR | Zbl | DOI

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) | MR | Zbl | DOI

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) | MR | Zbl | DOI

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 | MR | Zbl | DOI

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

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

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

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

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

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

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) | MR | Zbl | DOI

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) | MR | Zbl | mathdoc-id | DOI

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) | MR | Zbl | DOI

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

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) | MR | Zbl | DOI

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

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) | MR | Zbl | DOI

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) | MR | Zbl | mathdoc-id | DOI

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) | MR | Zbl | mathdoc-id

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

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 | MR | Zbl | DOI

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) | MR | Zbl | DOI

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

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

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) | MR | Zbl | DOI

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

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

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

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

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) | MR | Zbl | DOI

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) | MR | Zbl | DOI

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

Kollár, J., Astérisque, 211, 1992

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

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

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

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

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

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) | MR | Zbl | DOI

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

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) | MR | Zbl | mathdoc-id

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

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) | MR | Zbl | DOI

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

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) | MR | Zbl | DOI

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

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) | MR | Zbl | DOI

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) | MR | Zbl | mathdoc-id | DOI

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

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

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

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) | MR | Zbl | DOI

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) | MR | Zbl | DOI

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) | MR | Zbl | DOI

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) | MR | Zbl | DOI

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

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) | MR | Zbl | DOI

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