Advertisement

Mathematische Annalen

, Volume 373, Issue 1–2, pp 397–419 | Cite as

Kähler structures on spin 6-manifolds

  • Stefan SchreiederEmail author
  • Luca Tasin
Article

Abstract

We show that many spin 6-manifolds have the homotopy type but not the homeomorphism type of a Kähler manifold. Moreover, for given Betti numbers, there are only finitely many deformation types and hence topological types of smooth complex projective spin threefolds of general type. Finally, on a fixed spin 6-manifold, the Chern numbers take on only finitely many values on all possible Kähler structures.

Mathematics Subject Classification

Primary 14F45 32Q15 57R15 Secondary 14E30 57R20 

Notes

Acknowledgements

The first author is member of the SFB/TR 45. During parts of this project, the second author was supported by the DFG Emmy Noether-Nachwuchsgruppe “Gute Strukturen in der höherdimensionalen birationalen Geometrie” and thereby also member of the SFB/TR 45. We thank D. Kotschick for detailed comments and P. Cascini, M. Land, E. Sernesi, R. Svaldi and B. Totaro for conversations.

References

  1. 1.
    Amorós, J., Burger, M., Corlette, K., Kotschick, D., Toledo, D.: Fundamental Groups of Compact Kähler Manifolds, Mathematical Surveys and Monographs, vol. 44. AMS, Providence (1996)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bartels, A., Lück, W.: The Borel conjecture for hyperbolic and CAT(0)-groups. Ann. Math. 175, 631–689 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Campana, F., Höring, A., Peternell, T.: Abundance for Kähler threefolds. Ann. Sci. Éc. Norm. Supér. (4) 49(4), 971–1025 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cascini, P., Tasin, L.: On the Chern numbers of a smooth threefold. To appear in Trans. Am. Math. Soc. (2017)Google Scholar
  6. 6.
    Deligne, P., Griffiths, P., Morgan, J., Sullivan, D.: Real homotopy theory of Kähler manifolds. Invent. Math. 2(9), 245–274 (1975)CrossRefzbMATHGoogle Scholar
  7. 7.
    Friedman, R., Morgan, J.W.: On the diffeomorphism types of certain algebraic surfaces. I. J. Differ. Geom. 2(7), 297–369 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hacon, C.D., McKernan, J.: Boundedness of pluricanonical maps of varieties of general type. Invent. Math. 116, 1–25 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hirzebruch, F., Kodaira, K.: On the complex projective spaces. J. Math. Pures Appl. 3(6), 201–216 (1957)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Höring, A., Peternell, T.: Mori fibre spaces for Kähler threefolds. J. Math. Sci. Univ. Tokyo 22(1), 219–246 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Höring, A., Peternell, T.: Minimal models for Kähler threefolds. Invent. Math. 203(1), 217–264 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Iskovskikh, V.A., Prokhorov, Y.G.: Fano varieties, from: “Algebraic geometry, V”, (I.R. Shafarevich, editor). Encycl. Math. Sci. 47, 1–247 (1999). (Springer) Google Scholar
  13. 13.
    Kawamata, Y., Matsuki, K.: The number of the minimal models for a 3-fold of general type is finite. Math. Ann. 276(4), 595–598 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kollár, J.: Low degree polynomial equations: arithmetic, geometry and topology. European Congress of Mathematics, Vol. I (Budapest, 1996), 255–288, Progr. Math., vol. 168. Birkhäuser, Basel (1998)Google Scholar
  15. 15.
    Kollár, J.: Flops. Nagoya Math. J. 113, 15–36 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kollár, J., Mori, S.: Classification of three-dimensional flips. J. Am. Math. Soc. 5(3), 533–703 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kotschick, D.: Chern numbers and diffeomorphism types of projective varieties. J. Topol. 1, 518–526 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kotschick, D.: Characteristic numbers of algebraic varieties. Proc. Natl. Acad. Sci. USA 106, 10114–10115 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kotschick, D.: Topologically invariant Chern numbers of projective varieties. Adv. Math. 229, 1300–1312 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kotschick, D.: Updates on Hirzebruch’s 1954 problem list. arXiv:1305.4623
  21. 21.
    Kotschick, D., Schreieder, S.: The Hodge ring of Kähler manifolds. Compositio Math. 149, 637–657 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    LeBrun, C.: Topology versus Chern numbers of complex 3-folds. Pacific J. Math. 191, 123–131 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Libgober, A.S., Wood, J.W.: Uniqueness of the complex structure on Kähler manifolds of certain homotopy types. J. Differ. Geom. 3(2), 139–154 (1990)CrossRefzbMATHGoogle Scholar
  24. 24.
    Miyaoka, Y.: The Chern classes and Kodaira dimension of a minimal variety, Alg. Geom. Sendai 1985. Adv. Stud. Pure Math. 1, 449–476 (1987)CrossRefGoogle Scholar
  25. 25.
    Martinelli, D., Schreieder, S., Tasin, L.: On the number and boundedness of log minimal models of general type. arXiv:1610.08932
  26. 26.
    Mori, S.: Threefolds whose canonical bundles are not numerically effective. Ann. Math. 116, 133–176 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mori, S.: Flip theorem and the existence of minimal models for 3-folds. J. Am. Math. Soc. 1, 117–253 (1988)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mori, S., Prokhorov, Y.: On \({\cal{Q}}\)-conic bundles. Publ. Res. Inst. Math. Sci. 4(4), 315–369 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Novikov, S.P.: On manifolds with free abelian fundamental group and their application. Izv. Akad. Nauk SSSR Ser. Mat. 3, 207–246 (1966)MathSciNetGoogle Scholar
  30. 30.
    Okonek, C., Van de Ven, A.: Cubic forms and complex 3-folds. Enseign. Math. (2) 41(3–4), 297–333 (1995)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Schreieder, S., Tasin, L.: Algebraic structures with unbounded Chern numbers. J. Topol. 9, 849–860 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Schreieder, S.: On the construction problem for Hodge numbers. Geom. Topol. 1(9), 295–342 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Siu, Y.-T.: Invariance of plurigenera. Invent. Math. 134(3), 661–673 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Smale, S.: Diffeomorphisms of the 2-sphere. Proc. Am. Math. Soc. 1, 621–626 (1959)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Takayama, S.: Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165(3), 551–587 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Tsuji, H.: Stability of tangent bundles of minimal algebraic varieties. Topology 2(7), 429–442 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Tsuji, H.: Pluricanonical systems of projective varieties of general type. I. Osaka J. Math. 43(4), 967–995 (2006)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Voisin, C.: On the homotopy types of compact Kähler and complex projective manifolds. Invent. Math. 157, 329–343 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wall, C.T.C.: On the classification problem in differential topology, V. Invent. Math. 1, 355–374 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yau, S.-T.: Open problems in geometry. In: Yau, S.-T. (ed.) S.S. Chern: A Great Geometer of the Twentieth Century. International Press, Hong Kong (1992)Google Scholar
  41. 41.
    Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 7(4), 1798–1799 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhang, Y.: Miyaoka–Yau inequality for minimal projective manifolds of general type. Proc. Am. Math. Soc. 137, 2749–2754 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zhubr, A.V.: Classification of simply connected six-dimensional spinor manifolds, (English). Math. USSR Izv. 9(1975), 793–812 (1976)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Mathematisches Institut, LMU MünchenMünchenGermany
  2. 2.Mathematical Institute of the University of BonnBonnGermany

Personalised recommendations