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Singular foliations with trivial canonical class

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This paper describes the structure of singular codimension one foliations with numerically trivial canonical bundle on complex projective manifolds.

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References

  1. Araujo, C., Druel, S., Kovács, S.: Cohomological characterizations of projective spaces and hyperquadrics. Invent. Math. 174(2), 233–253 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araujo, C., Druel, S.: On Fano foliations. Adv. Math. 238, 70–118 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baum, P., Bott, R.: Singularities of holomorphic foliations. J. Differ. Geom. 7, 279–342 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beauville, A.: Variétés kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18(4), 755–782 (1983)

    Article  MATH  Google Scholar 

  5. Bogomolov, F.: The decomposition of Kähler manifolds with a trivial canonical class. Math. USSR-Sbornik 22(4), 580–583 (1974)

    Article  MATH  Google Scholar 

  6. Bogomolov, F., McQuillan, M.: Rational Curves on Foliated Varieties. Foliation theory in algebraic geometry. Springer, Cham (2016)

    Book  MATH  Google Scholar 

  7. Boucksom, S., Demailly, J.-P., Paun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebr. Geom. 22(2), 201–248 (2013)

    Article  MATH  Google Scholar 

  8. Bravo, A., Encinas, S., Villamayor, O.: A simplified proof of desingularization and applications. Rev. Mat. Iberoamericana 21(2), 349–458 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brion, M.: Some basic results on actions of nonaffine algebraic groups. In: Campbell, H., Helminck, A., Kraft, H., Wehlau, D. (eds.) Symmetry and Spaces. Progress in Mathematics, vol. 278, pp. 1–20. Birkhaüser, Boston (2010)

    Google Scholar 

  10. Brunella, M.: Birational Geometry of Foliations. Publicaçoes Matemáticas do IMPA. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2004)

  11. Brunella, M., Mendes, L.G.: Bounding the degree of solutions to Pfaff equations. Publ. Mat. 44(2), 593–604 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brunella, M., Pereira, J.V., Touzet, F.: Kähler manifolds with split tangent bundle. Bull. Soc. Math. France 134(2), 241–252 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Campana, F., Păun, M.: Foliations with positive slopes and birational stability of orbifold cotangent bundles. (2015) arXiv:1508.02456 [math.AG]

  14. Campana, F., Peternell, T.: Geometric stability of the cotangent bundle and the universal cover of a projective manifold (with an appendix by Matei Toma). Bull. Soc. Math. France 139, 41–74 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Casale, G.: Suites de Godbillon–Vey et intégrales premières. C. R. Math. Acad. Sci. Paris 335, 1003–1006 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cerveau, D., Lins Neto, A.: Irreducible components of the space of holomorphic foliations of degree two in \(\mathbf{C}{\rm P}(n)\), \(n\ge 3\). Ann. Math. 143, 577–612 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cerveau, D., Lins Neto, A., Loray, F., Pereira, J.V., Touzet, F.: Algebraic reduction theorem for complex codimension one singular foliations. Comment. Math. Helv. 81, 157–169 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cerveau, D., Lins Neto, A., Loray, F., Pereira, J.V., Touzet, F.: Complex codimension one singular foliations and Godbillon–Vey sequences. Moscow Math. J. 7, 21–54 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Coutinho, S.C., Pereira, J.V.: On the density of algebraic foliations without algebraic invariant sets. J. Reine Angew. Math. 594, 117–135 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001). xiv+233 pp

    Book  MATH  Google Scholar 

  21. Demailly, J.-P.: On the Frobenius Integrability of Certain Holomorphic \(p\)-forms. Complex geometry (Gottingen, 2000). Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  22. Druel, S.: Structures de Poisson sur les variétés algébriques de dimension \(3\). Bull. Soc. Math. France 127(2), 229–253 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Druel, S.: Some remarks on regular foliations with numerically trivial canonical class. EPIGA 1 (2017)

  24. Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. AMS 16(5), 57–67 (2003)

    MathSciNet  MATH  Google Scholar 

  25. Hartshorne, R.: Ample vector bundles on curves. Nagoya Math. J. 43, 73–89 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hwang, J.-H., Viehweg, E.: Characteristic foliation on a hypersurface of general type in a projective symplectic manifold. Compos. Math. 146(2), 497–506 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Iitaka, S.: Logarithmic forms of algebraic varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(3), 525–544 (1976)

    MathSciNet  MATH  Google Scholar 

  28. Jacobson, N.: Lie Algebras, p. 1979. Dover Publications Inc, New York (1962). (Republication of the 1962 original)

    Google Scholar 

  29. Kebekus, S., Solá Conde, L., Toma, M.: Rationally connected foliations after Bogomolov and McQuillan. J. Algebr. Geom. 16(1), 65–81 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kollár, J., et al.: Flips and abundance for algebraic threefolds. Astérisque 211, 103–114 (1993)

    Google Scholar 

  31. Kollár, J.: Singularities of the Minimal Model Program. Cambridge tracts in mathematics, vol. 200. Cambridge University Press, Cambridge (2013)

    Book  MATH  Google Scholar 

  32. Kollár, J.: Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of modern surveys in mathematics, vol. 32. Springer, Berlin (1996)

  33. Kollár, J., Larsen, M.: Quotients of Calabi–Yau varieties. Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 179–211, Progr. Math., vol. 270 (2009)

  34. Kupka, I.: The singularities of integrable structurally stable Pfaffian forms. Proc. Nat. Acad. Sci. USA 52, 1431–1432 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lima, R., Pereira, J.V.: A characterization of diagonal Poisson structures. Bull. Lond. Math. Soc. 46(6), 1203–1217 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Loray, F.: Pseudo-groupe d’une singularité de feuilletage holomorphe en dimension deux. HAL:hal-00016434, version 1. https://hal.archives-ouvertes.fr/hal-00016434v1

  37. Loray, F., Pereira, J.V.: Transversely projective foliations on surfaces: existence of normal forms and prescription of the monodromy. Intern. J. Math. 18, 723–747 (2007)

    Article  MATH  Google Scholar 

  38. Loray, F., Pereira, J.V., Touzet, F.: Foliations with trivial canonical bundle on Fano \(3\)-folds. Math. Nachr. 286(8–9), 921–940 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. McQuillan, M.: Canonical models of foliations. Pure Appl. Math. Q. 4(3), 877–1012 (2008). part 2

    Article  MathSciNet  MATH  Google Scholar 

  40. McQuillan, M., Panazzolo, D.: Almost étale resolution of foliations. J. Differ. Geom. 95(2), 279–319 (2013)

    Article  MATH  Google Scholar 

  41. Mehta, V.B., Ramanathan, A.: Semistable sheaves on projective varieties and their restriction to curves. Math. Ann. 258(3), 213–224 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  42. Miyaoka, Y.: Deformations of a Morphism Along a Foliation and Applications. Algebraic geometry, Bowdoin, 1985. In: Proceedings of Symposis in Pure Mathematics, Part 1, vol. 46, pp. 245–268. American Mathatematical Society, Providence (1987)

  43. Miyaoka, Y., Peternell, T.: Geometry of Higher Dimensional Algebraic Varieties. DMV seminar, vol. 26. Birkhäuser, Basel (1997)

    Book  MATH  Google Scholar 

  44. Neumann, S.: A decomposition of the Moving cone of a projective manifold according to the Harder–Narasimhan filtration of the tangent bundle. Ph.D. thesis. Universitat Freiburg, (2009). http://www.freidok.uni-freiburg.de/volltexte/7287/pdf/Diss_Neumann.pdf

  45. Pereira, J.V., Pirio, L.: The classification of exceptional CDQL webs on compact complex surfaces. Int. Math. Res. Not. IMRN 12, 2169–2282 (2010)

    MathSciNet  MATH  Google Scholar 

  46. Peternell, T.: Minimal varieties with trivial canonical classes. I. Math. Z. 217(3), 377–405 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  47. Peternell, T.: Generically nef vector bundles and geometric applications. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds.) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol. 8, pp. 345–368. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  48. Polishchuk, A.: Algebraic geometry of Poisson brackets. Algebraic geometry, 7. J. Math. Sci. 84(5), 1413–1444 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  49. Scardua, B.: Transversely affine and transversely projective holomorphic foliations. Ann. Sci. École Norm. Sup. (4) 30(2), 169–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  50. Shepherd-Barron, N.I.: Semi-stability and reduction mod \(p\). Topology 37(3), 659–664 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  51. Simpson, C.: Subspaces of moduli spaces of rank one local systems. Ann. Sci. École Norm. Sup. (4) 26(3), 361–401 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sommese, A.J.: Holomorphic vector-fields on compact Kähler manifolds. Math. Ann. 210, 75–82 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  53. Touzet, F.: Feuilletages holomorphes de codimension un dont la classe canonique est triviale. Ann. Sci. Éc. Norm. Supér. (4) 41(4), 655–668 (2008)

    MathSciNet  Google Scholar 

  54. Touzet, F.: Uniformisation de l’espace des feuilles de certains feuilletages de codimension un. Bull. Braz. Math. Soc. (N.S.) 44(3), 351–391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We are very grateful to Stéphane Druel for useful discussions and for bringing [32, Theorem VI.1.3] to our knowledge. This paper also owns a lot to Michael McQuillan who caught a number of mistakes in previous versions, called our attention to the relevance of foliated canonical singularities to our study, and made a number of other thoughtful suggestions. We also thank the anonymous referee for pointing out some inaccuracies. J. V. Pereira was partially support by Cnpq and FAPERJ. Finally, we acknowledge financial support from ANR-16-CE40-0008 project Foliage and Brazilian-French Network in Mathematics.

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Authors and Affiliations

  1. CNRS, IRMAR - UMR 6625, Univ Rennes, 35000, Rennes, France

    Frank Loray & Frédéric Touzet

  2. IMPA, Estrada Dona Castorina, 110, Horto, Rio de Janeiro, Brazil

    Jorge Vitório Pereira

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  1. Frank Loray
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Correspondence to Frédéric Touzet.

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Loray, F., Pereira, J.V. & Touzet, F. Singular foliations with trivial canonical class. Invent. math. 213, 1327–1380 (2018). https://doi.org/10.1007/s00222-018-0806-0

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