Projectively flat klt varieties

Daniel Greb; Stefan Kebekus; Thomas Peternell

doi:10.5802/jep.164

Projectively flat klt varieties
[Variétés klt projectivement plates]

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

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

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

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2021__8__1005_0,
     author = {Daniel Greb and Stefan Kebekus and Thomas Peternell},
     title = {Projectively flat klt varieties},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1005--1036},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.164},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.164/}
}

TY  - JOUR
AU  - Daniel Greb
AU  - Stefan Kebekus
AU  - Thomas Peternell
TI  - Projectively flat klt varieties
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2021
SP  - 1005
EP  - 1036
VL  - 8
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.164/
DO  - 10.5802/jep.164
LA  - en
ID  - JEP_2021__8__1005_0
ER  -

%0 Journal Article
%A Daniel Greb
%A Stefan Kebekus
%A Thomas Peternell
%T Projectively flat klt varieties
%J Journal de l’École polytechnique — Mathématiques
%D 2021
%P 1005-1036
%V 8
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.164/
%R 10.5802/jep.164
%G en
%F JEP_2021__8__1005_0

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

Bibliographie
Cité par

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

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

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

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

[BB70] P. F. Baum & R. Bott - “On the zeros of meromorphic vector-fields”, in Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, p. 29-47 | DOI | Zbl

[BC15] C. Birkar & J. A. Chen - “Varieties fibred over abelian varieties with fibres of log general type”, Adv. Math. 270 (2015), p. 206-222 | DOI | MR | Zbl

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

[Bea00] A. Beauville - “Complex manifolds with split tangent bundle”, in Complex analysis and algebraic geometry, de Gruyter, Berlin, 2000, p. 61-70 | DOI | Zbl

[BHPVdV04] W. P. Barth, K. Hulek, C. A. M. Peters & A. Van de Ven - Compact complex surfaces, Ergeb. Math. Grenzgeb. (3), vol. 4, Springer-Verlag, Berlin, 2004 | DOI | MR | Zbl

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

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

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

[CCE15] F. Campana, B. Claudon & P. Eyssidieux - “Représentations linéaires des groupes kählériens: factorisations et conjecture de Shafarevich linéaire”, Compositio Math. 151 (2015) no. 2, p. 351-376 | DOI | Zbl

[CKP12] F. Campana, V. Koziarz & M. Păun - “Numerical character of the effectivity of adjoint line bundles”, Ann. Inst. Fourier (Grenoble) 62 (2012) no. 1, p. 107-119 | DOI | Numdam | MR | Zbl

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

[Dem02] J.-P. Demailly - “On the Frobenius integrability of certain holomorphic $p$ -forms”, in Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, p. 93-98 | DOI | Zbl

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

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

[FG12] O. Fujino & Y. Gongyo - “On canonical bundle formulas and subadjunctions”, Michigan Math. J. 61 (2012) no. 2, p. 255-264 | DOI | MR | Zbl

[FL81] W. Fulton & R. Lazarsfeld - “Connectivity and its applications in algebraic geometry”, in Algebraic geometry (Chicago, Ill., 1980), Lect. Notes in Math., vol. 862, Springer, Berlin, 1981, p. 26-92 | DOI | MR | Zbl

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

[GKK10] D. Greb, S. Kebekus & S. J. Kovács - “Extension theorems for differential forms, and Bogomolov-Sommese vanishing on log canonical varieties”, Compositio Math. 146 (2010), p. 193-219, A slightly extended version is available as arXiv:0808.3647 | DOI | MR | Zbl

[GKKP11] D. Greb, S. Kebekus, S. J. Kovács & T. Peternell - “Differential forms on log canonical spaces”, Publ. Math. Inst. Hautes Études Sci. 114 (2011) no. 1, p. 87-169, An extended version with additional graphics is available as arXiv:1003.2913 | DOI | Numdam | MR | Zbl

[GKP16] D. Greb, S. Kebekus & T. Peternell - “Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties”, Duke Math. J. 165 (2016) no. 10, p. 1965-2004 | DOI | Zbl

[GKP20] D. Greb, S. Kebekus & T. Peternell - “Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor”, 2020 | arXiv

[GKPT19a] D. Greb, S. Kebekus, T. Peternell & B. Taji - “The Miyaoka-Yau inequality and uniformisation of canonical models”, Ann. Sci. École Norm. Sup. (4) 52 (2019) no. 6, p. 1487-1535 | DOI | MR

[GKPT19b] D. Greb, S. Kebekus, T. Peternell & B. Taji - “Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles”, Compositio Math. 155 (2019) no. 2, p. 289-323 | DOI | MR | Zbl

[GKPT20] D. Greb, S. Kebekus, T. Peternell & B. Taji - “Harmonic metrics on Higgs sheaves and uniformization of varieties of general type”, Math. Ann. 378 (2020) no. 3-4, p. 1061-1094 | DOI | MR | Zbl

[GKT18] D. Greb, S. Kebekus & B. Taji - “Uniformisation of higher-dimensional varieties”, in Algebraic Geometry (Salt Lake City, 2015) (T. de Fernex, B. Hassett, M. Mustaţă, M. Olsson, M. Popa & R. Thomas, eds.), Proc. Sympos. Pure Math., vol. 97, American Mathematical Society, Providence, RI, 2018, p. 277-308 | DOI | MR

[Gro61] A. Grothendieck - “Techniques de construction en géométrie analytique. V. Fibrés vectoriels, fibrés projectifs, fibrés en drapeaux”, in Séminaire Henri Cartan, vol. 13, no. 1, Secrétariat mathématique, Paris, 1960–1961, p. 1-15, Exp. no. 12

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

[HM07] C. D. Hacon & J. McKernan - “On Shokurov’s rational connectedness conjecture”, Duke Math. J. 138 (2007) no. 1, p. 119-136 | DOI | MR | Zbl

[HO84] A. Huckleberry & E. Oeljeklaus - Classification theorems for almost homogeneous spaces, vol. 9, Université de Nancy, Institut Élie Cartan, Nancy, 1984 | MR | Zbl

[JR13] P. Jahnke & I. Radloff - “Semistability of restricted tangent bundles and a question of I. Biswas”, Internat. J. Math. 24 (2013) no. 1, article ID 1250122, 15 pages | DOI | MR | Zbl

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

[Kaw85b] Y. Kawamata - “Pluricanonical systems on minimal algebraic varieties”, Invent. Math. 79 (1985) no. 3, p. 567-588 | DOI | MR | Zbl

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

[KMM87] Y. Kawamata, K. Matsuda & K. Matsuki - “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

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

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

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

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

[Lai11] C.-J. Lai - “Varieties fibered by good minimal models”, Math. Ann. 350 (2011) no. 3, p. 533-547 | DOI | MR | Zbl

[LT18] S. Lu & B. Taji - “A characterization of finite quotients of abelian varieties”, Internat. Math. Res. Notices (2018) no. 1, p. 292-319 | DOI | MR | Zbl

[Nak98] N. Nakayama - “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] N. Nakayama - Zariski-decomposition and abundance, MSJ Memoirs, vol. 14, Mathematical Society of Japan, Tokyo, 2004 | DOI | MR | Zbl

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

[Ros68] H. Rossi - “Picard variety of an isolated singular point”, Rice Univ. Studies 54 (1968) no. 4, p. 63-73 | MR | Zbl

[Sch16] M. Schwald - “Low degree Hodge theory for klt varieties”, 2016 | arXiv

[ST71] Y.-T. Siu & G. Trautmann - Gap-sheaves and extension of coherent analytic subsheaves, Lect. Notes in Math., vol. 172, Springer-Verlag, Berlin, 1971 | DOI | MR | Zbl

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

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

Cité par Sources :