Projectively flat klt varieties

Greb, Daniel; Kebekus, Stefan; Peternell, Thomas

doi:10.5802/jep.164

Projectively flat klt varieties
[Variétés klt projectivement plates]

Greb, Daniel ¹ ; Kebekus, Stefan ^2,³ ; Peternell, Thomas ⁴

¹ Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen 45117 Essen, Germany
² Mathematisches Institut, Albert-Ludwigs-Universität Freiburg Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany
³ & Freiburg Institute for Advanced Studies (FRIAS) Freiburg im Breisgau, Germany
⁴ Mathematisches Institut, Universität Bayreuth 95440 Bayreuth, Germany

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1005-1036.

Résumé
Abstract

Dans le cadre des problèmes d’uniformisation, nous étudions les variétés projectives avec singularités klt dont le faisceau cotangent admet une structure projective plate sur le lieu lisse. En généralisant le travail de Jahnke-Radloff, nous montrons que les quotients des tores sont les seules variétés klt avec un faisceau cotangent semi-stable et des classes de Chern extrémales. Un résultat analogue pour les variétés avec un faisceau cotangent normalisé nef s’ensuit.

In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows.

Reçu le : 2020-10-20
Accepté le : 2021-04-30
Publié le : 2021-05-20

DOI : 10.5802/jep.164

Classification : 32Q30, 32Q26, 14E20, 14E30, 53B10
Keywords: Bogomolov-Gieseker inequality, Abelian variety, klt singularities, Miyaoka-Yau inequality, stability, projective flatness, uniformisation
Mot clés : Inégalité de Bogomolov-Gieseker, variété abélienne, singularités klt, inégalité de Miyaoka-Yau, stabilité, platitude projective, uniformisation

Affiliations des auteurs :

Greb, Daniel ¹ ; Kebekus, Stefan ^{2, 3} ; Peternell, Thomas ⁴

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Greb, Daniel; Kebekus, Stefan; Peternell, Thomas. Projectively flat klt varieties. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 1005-1036. doi : 10.5802/jep.164. http://www.numdam.org/articles/10.5802/jep.164/

Bibliographie
Cité par

[Akh95] Akhiezer, Dmitri N. Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995 | DOI | MR | Zbl

[Alp87] Alperin, Roger C. An elementary account of Selberg’s lemma, Enseign. Math. (2), Volume 33 (1987) no. 3-4, pp. 269-273 | DOI | MR | Zbl

[Amb05] Ambro, Florin The moduli $b$ -divisor of an lc-trivial fibration, Compositio Math., Volume 141 (2005) no. 2, pp. 385-403 | DOI | MR | Zbl

[Anc82] Ancona, Vincenzo Faisceaux amples sur les espaces analytiques, Trans. Amer. Math. Soc., Volume 274 (1982) no. 1, pp. 89-100 | DOI | MR | Zbl

[BB70] Baum, Paul F.; Bott, Raoul On the zeros of meromorphic vector-fields, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 29-47 | DOI | Zbl

[BC15] Birkar, Caucher; Chen, Jungkai Alfred Varieties fibred over abelian varieties with fibres of log general type, Adv. Math., Volume 270 (2015), pp. 206-222 | DOI | MR | Zbl

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

[Bea00] Beauville, Arnaud Complex manifolds with split tangent bundle, Complex analysis and algebraic geometry, de Gruyter, Berlin, 2000, pp. 61-70 | DOI | Zbl

[BHPVdV04] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), 4, Springer-Verlag, Berlin, 2004 | DOI | MR | Zbl

[BS76] Bănică, Constantin; Stănăşilă, Octavian Algebraic methods in the global theory of complex spaces, Editura Academiei, Bucharest; John Wiley & Sons, London-New York-Sydney, 1976, 296 pages | Zbl

[Cam91] Campana, Frédéric On twistor spaces of the class $𝒞$ , J. Differential Geom., Volume 33 (1991) no. 2, pp. 541-549 | DOI | MR

[Car57] Cartan, Henri Quotient d’un espace analytique par un groupe d’automorphismes, Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton University Press, Princeton, NJ, 1957, pp. 90-102 | DOI | Zbl

[CCE15] Campana, Fréderic; Claudon, Benoît; Eyssidieux, Philippe Représentations linéaires des groupes kählériens: factorisations et conjecture de Shafarevich linéaire, Compositio Math., Volume 151 (2015) no. 2, pp. 351-376 | DOI | Zbl

[CKP12] Campana, Frédéric; Koziarz, Vincent; Păun, Mihai Numerical character of the effectivity of adjoint line bundles, Ann. Inst. Fourier (Grenoble), Volume 62 (2012) no. 1, pp. 107-119 | DOI | Numdam | MR | Zbl

[Deb01] Debarre, Olivier Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001 | DOI | Zbl

[Dem02] Demailly, Jean-Pierre On the Frobenius integrability of certain holomorphic $p$ -forms, Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 93-98 | DOI | Zbl

[Dru00] Druel, Stéphane Variétés algébriques dont le fibré tangent est totalement décomposé, J. reine angew. Math., Volume 522 (2000), pp. 161-171 | DOI | MR | Zbl

[Dru14] Druel, Stéphane The Zariski-Lipman conjecture for log canonical spaces, Bull. London Math. Soc., Volume 46 (2014) no. 4, pp. 827-835 | DOI | MR | Zbl

[FG12] Fujino, Osamu; Gongyo, Yoshinori On canonical bundle formulas and subadjunctions, Michigan Math. J., Volume 61 (2012) no. 2, pp. 255-264 | DOI | MR | Zbl

[FL81] Fulton, William; Lazarsfeld, Robert Connectivity and its applications in algebraic geometry, Algebraic geometry (Chicago, Ill., 1980) (Lect. Notes in Math.), Volume 862, Springer, Berlin, 1981, pp. 26-92 | DOI | MR | Zbl

[Fuj13] Fujino, Osamu On maximal Albanese dimensional varieties, Proc. Japan Acad. Ser. A Math. Sci., Volume 89 (2013) no. 8, pp. 92-95 | DOI | MR | Zbl

[GKK10] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J. Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties, Compositio Math., Volume 146 (2010), pp. 193-219 (A slightly extended version is available as arXiv:0808.3647) | DOI | MR | Zbl

[GKKP11] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor J.; Peternell, Thomas Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci., Volume 114 (2011) no. 1, pp. 87-169 (An extended version with additional graphics is available as arXiv:1003.2913) | DOI | Numdam | MR | Zbl

[GKP16] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J., Volume 165 (2016) no. 10, pp. 1965-2004 | DOI | Zbl

[GKP20] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor, 2020 | arXiv

[GKPT19a] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas; Taji, Behrouz The Miyaoka-Yau inequality and uniformisation of canonical models, Ann. Sci. École Norm. Sup. (4), Volume 52 (2019) no. 6, pp. 1487-1535 | DOI | MR

[GKPT19b] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas; Taji, Behrouz Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles, Compositio Math., Volume 155 (2019) no. 2, pp. 289-323 | DOI | MR | Zbl

[GKPT20] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas; Taji, Behrouz Harmonic metrics on Higgs sheaves and uniformization of varieties of general type, Math. Ann., Volume 378 (2020) no. 3-4, pp. 1061-1094 | DOI | MR | Zbl

[GKT18] Greb, Daniel; Kebekus, Stefan; Taji, Behrouz Uniformisation of higher-dimensional varieties, Algebraic Geometry (Salt Lake City, 2015) (de Fernex, Tommaso; Hassett, Brendan; Mustaţă, Mircea; Olsson, Martin; Popa, Mihnea; Thomas, Richard, eds.) (Proc. Sympos. Pure Math.), Volume 97, American Mathematical Society, Providence, RI, 2018, pp. 277-308 | DOI | MR

[Gro61] Grothendieck, Alexandre Techniques de construction en géométrie analytique. V. Fibrés vectoriels, fibrés projectifs, fibrés en drapeaux, Séminaire Henri Cartan, Volume 13, no. 1, Secrétariat mathématique, Paris, 1960–1961, pp. 1-15 (Exp. no. 12)

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

[HM07] Hacon, Christopher D.; McKernan, James On Shokurov’s rational connectedness conjecture, Duke Math. J., Volume 138 (2007) no. 1, pp. 119-136 | DOI | MR | Zbl

[HO84] Huckleberry, Alan; Oeljeklaus, Eberhard Classification theorems for almost homogeneous spaces, 9, Université de Nancy, Institut Élie Cartan, Nancy, 1984 | MR | Zbl

[JR13] Jahnke, Priska; Radloff, Ivo Semistability of restricted tangent bundles and a question of I. Biswas, Internat. J. Math., Volume 24 (2013) no. 1, 1250122, 15 pages | DOI | MR | Zbl

[Kaw85a] Kawamata, Yujiro Minimal models and the Kodaira dimension of algebraic fiber spaces, J. reine angew. Math., Volume 363 (1985), pp. 1-46 | DOI | MR | Zbl

[Kaw85b] Kawamata, Yujiro Pluricanonical systems on minimal algebraic varieties, Invent. Math., Volume 79 (1985) no. 3, pp. 567-588 | DOI | MR | Zbl

[KM98] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge, 1998 | DOI | MR | Zbl

[KMM87] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji 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

[Kob87] Kobayashi, Shoshichi Differential geometry of complex vector bundles, Publications of the Math. Society of Japan, 15, Iwanami Shoten and Princeton University Press, Princeton, NJ, 1987 | MR | Zbl

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

[Kol95] Kollár, János Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1995 | DOI | Zbl

[KS21] Kebekus, Stefan; Schnell, Christian Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities, J. Amer. Math. Soc., Volume 34 (2021) no. 2, p. 315-–368 | DOI

[Lai11] Lai, Ching-Jui Varieties fibered by good minimal models, Math. Ann., Volume 350 (2011) no. 3, pp. 533-547 | DOI | MR | Zbl

[LT18] Lu, Steven; Taji, Behrouz A characterization of finite quotients of abelian varieties, Internat. Math. Res. Notices (2018) no. 1, pp. 292-319 | DOI | MR | Zbl

[Nak98] Nakayama, Noboru Normalized tautological divisors of semi-stable vector bundles (1998) (RIMS preprint 1214, available at https://www.kurims.kyoto-u.ac.jp/preprint/preprint_y1998.html)

[Nak04] Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, Tokyo, 2004 | DOI | MR | Zbl

[Ray70] Raynaud, Michel Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lect. Notes in Math., 119, Springer-Verlag, Berlin-New York, 1970 | DOI | Zbl

[Ros68] Rossi, Hugo Picard variety of an isolated singular point, Rice Univ. Studies, Volume 54 (1968) no. 4, pp. 63-73 https://scholarship.rice.edu/handle/1911/62964 | MR | Zbl

[Sch16] Schwald, Martin Low degree Hodge theory for klt varieties, 2016 | arXiv

[ST71] Siu, Yum-Tong; Trautmann, Günther Gap-sheaves and extension of coherent analytic subsheaves, Lect. Notes in Math., 172, Springer-Verlag, Berlin, 1971 | DOI | MR | 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

[Vie82] Viehweg, Eckart Die Additivität der Kodaira Dimension für projektive Faserräume über Varietäten des allgemeinen Typs, J. reine angew. Math., Volume 330 (1982), pp. 132-142 | DOI | Zbl

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