Advertisement

Algebraic integrability of foliations with numerically trivial canonical bundle

  • Andreas HöringEmail author
  • Thomas Peternell
Article
  • 70 Downloads

Abstract

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent works of Druel and Greb–Guenancia–Kebekus this establishes the Beauville–Bogomolov decomposition for minimal models with trivial canonical class.

Mathematics Subject Classification

14J32 37F75 14E30 

Notes

Acknowledgements

We thank S. Cantat and P. Graf for some very useful references. This work was partially supported by the Agence Nationale de la Recherche grant project Foliage (ANR-16-CE40-0008) and by the DFG project “Zur Positivität in der komplexen Geometrie”. We thank the referees for numerous suggestions to improve this text.

References

  1. 1.
    Boucksom, S., Demailly, J.-P., Păun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebr. Geom. 22, 201–248 (2013)CrossRefzbMATHGoogle Scholar
  2. 2.
    Beauville, A.: Variétés Kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1984)CrossRefzbMATHGoogle Scholar
  3. 3.
    Balaji, V., Kollár, J.: Holonomy groups of stable vector bundles. Publ. Res. Inst. Math. Sci. 44(2), 183–211 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bost, J.-B.: Algebraic leaves of algebraic foliations over number fields. Publ. Math. Inst. Hautes Études Sci. 93, 161–221 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bando, S., Siu, Y.-T.: Stable sheaves and Einstein–Hermitian metrics. In: Mabuchi, T., Noguchi, J., Ochiai, T. (eds.) Geometry and Analysis on Complex Manifolds, pp. 39–50. World Sci. Publ, River Edge (1994)CrossRefGoogle Scholar
  6. 6.
    Cao, J., Höring, A.: A decomposition theorem for projective manifolds with nef anticanonical bundle. arXiv preprint arXiv: 1706.08814, to appear in Journal of Algebraic Geometry (2017)
  7. 7.
    Campana, F., Păun, M.: Foliations with positive slopes and birational stability of orbifold cotangent bundles. arXiv preprint arXiv: 1508.02456 (2015)
  8. 8.
    Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  9. 9.
    Demailly, J.-P.: A numerical criterion for very ample line bundles. J. Differ. Geom. 37(2), 323–374 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Druel, S., Guenancia, H.: A decomposition theorem for smoothable varieties with trivial canonical class. J. Éc. Polytech. Math. 5, 117–147 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demailly, J.-P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebr. Geom. 3(2), 295–345 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Druel, S.: A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. Invent. Math. 211(1), 245–296 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ein, L., Lazarsfeld, R., Mustata, M., Nakamaye, M., Popa, M.: Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56(6), 1701–1734 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fulton, W., Harris, J.: Representation Theory: A First Course, Readings in Mathematics. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)zbMATHGoogle Scholar
  15. 15.
    Fulger, M., Lehmann, B.: Zariski decompositions of numerical cycle classes. J. Algebr. Geom. 26(1), 43–106 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Flenner, H.: Restrictions of semistable bundles on projective varieties. Comment. Math. Helv. 59(4), 635–650 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Greb, D., Guenancia, H., Kebekus, S.: Klt varieties with trivial canonical class. holonomy, differential forms, and fundamental groups. arXiv preprint arXiv: 1704.01408, to appear in Geometry & Topology (2017)
  18. 18.
    Greb, D., Kebekus, S., Kovács, S.J., Peternell, T.: Differential forms on log canonical spaces. Publ. Math. Inst. Hautes Études Sci. 114, 87–169 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Greb, D., Kebekus, S., Peternell, T.: Etale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties. Duke Math. J. 165(10), 1965–2004 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Greb, D., Kebekus, S., Peternell, T.: Movable curves and semistable sheaves. Int. Math. Res. Not. IMRN 2, 536–570 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Greb, D., Kebekus, S., Peternell, T.: Singular spaces with trivial canonical class. In: Kollár, J., Fujino, O., Mukai, S., Nakayama, N. (eds.) Minimal Models and Extremal Rays (Kyoto, 2011). Volume 70 of Adv. Stud. Pure Math., pp. 67–113. Math. Soc. Japan, Tokyo (2016)CrossRefGoogle Scholar
  22. 22.
    Hartshorne, R.: Ample Subvarieties of Algebraic Varieties. Notes Written in Collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156. Springer, Berlin (1970)CrossRefzbMATHGoogle Scholar
  23. 23.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York (1977)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties, Volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1998). With the collaboration of C. H. Clemens and A. CortiGoogle Scholar
  25. 25.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Volume 15 of Publications of the Mathematical Society of Japan. Kanô Memorial Lectures, 5. Princeton University Press, Princeton (1987)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kubota, K.: Ample sheaves. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 17, 421–430 (1970)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Langer, A.: Semistable sheaves in positive characteristic. Ann. Math. (2) 159(1), 251–276 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lazarsfeld, R.: Positivity in Algebraic Geometry. I: Classical Setting: Line Bundles and Linear Series. Volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (2004)zbMATHGoogle Scholar
  29. 29.
    Lazarsfeld, R.: Positivity in Algebraic Geometry. II: Positivity for Vector Bundles, and Multiplier Ideals. Volume 49 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  30. 30.
    Loray, F., Pereira, J.V., Touzet, F.: Foliations with trivial canonical bundle on Fano 3-folds. Math. Nachr. 286(8–9), 921–940 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Loray, F., Pereira, J.V., Touzet, F.: Singular foliations with trivial canonical class. Invent. Math. 213(3), 1327–1380 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lübke, M., Teleman, A.: The Kobayashi–Hitchin Correspondence. World Scientific Publishing Co. Inc, River Edge (1995)CrossRefzbMATHGoogle Scholar
  33. 33.
    Lu, S., Taji, B.: A characterization of finite quotients of abelian varieties. Int. Math. Res. Not. IMRN 2018(1), 292–319 (2018)MathSciNetGoogle Scholar
  34. 34.
    Mistretta, E.: Holomorphic symmetric differentials and a birational characterization of abelian varieties. arXiv preprint arXiv: 1808.00865 (2018)
  35. 35.
    Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety. In: Oda, T. (ed.) Algebraic Geometry, Sendai, 1985. Volume 10 of Adv. Stud. Pure Math., pp. 449–479. North-Holland, Amsterdam (1987)CrossRefGoogle Scholar
  36. 36.
    Miyaoka, Y., Peternell, T.: Geometry of Higher-Dimensional Algebraic Varieties. DMV Seminar, Vol. 26. Birkhäuser Verlag, Basel (1997)CrossRefzbMATHGoogle Scholar
  37. 37.
    Mehta, V.B., Ramanathan, A.: Restriction of stable sheaves and representations of the fundamental group. Invent. Math. 77(1), 163–172 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mehta, V.B., Ramanathan, A.: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258(3), 213–224 (1981/82)Google Scholar
  39. 39.
    Nakayama, N.: Normalized tautological divisors of semi-stable vector bundles. Surikaisekikenkyusho Kokyuroku, (1078), pp 167–173 (1999). Free resolutions of coordinate rings of projective varieties and related topics (Japanese) (Kyoto, 1998)Google Scholar
  40. 40.
    Nakayama, N.: Zariski-Decomposition and Abundance. MSJ Memoirs, Vol. 14. Mathematical Society of Japan, Tokyo (2004)CrossRefzbMATHGoogle Scholar
  41. 41.
    Paun, M.: Sur l’effectivité numérique des images inverses de fibrés en droites. Math. Ann. 310(3), 411–421 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Peternell, T.: Minimal varieties with trivial canonical classes. I. Math. Z. 217(3), 377–405 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Pereira, J.V., Touzet, F.: Foliations with vanishing Chern classes. Bull. Braz. Math. Soc. (N.S.) 44(4), 731–754 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Simpson, C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Touzet, F.: Feuilletages holomorphes de codimension un dont la classe canonique est triviale. Ann. Sci. Éc. Norm. Supér. (4) 41(4), 655–668 (2008)MathSciNetGoogle Scholar
  46. 46.
    Weyl, H.: The Classical Groups: Their Invariants and Representations. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1949). Princeton Paperbacks zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CNRS, LJADUniversité Côte d’AzurNiceFrance
  2. 2.Mathematisches InstitutUniversität BayreuthBayreuthGermany

Personalised recommendations