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Odd Type Generalized Complex Structures on 4-Manifolds

  • Haojie Chen
  • Xiaolan NieEmail author
Article
  • 8 Downloads

Abstract

We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. We also obtain an equivalent description of generalized complex structures in terms of almost bihermitian structures.

Notes

Acknowledgements

We would like to thank Professor Tian-Jun Li for the support, encouragement, and many useful discussions, especially for pointing out Corollary 1.3. Also, we like to thank Professors Tedi Draghici, Weiwei Wu for the discussions and Professor Marco Gualtieri for helpful communications. The authors thank the anonymous referees whose comments and suggestions helped improve the paper. The first-named author is partially supported by Zhejiang Provincial Natural Science Foundation (Grant No. LY19A010017). The second-named author is partially supported by National Natural Science Foundation of China (Grant No. 11801516).

References

  1. 1.
    Apostolov, V., Gualtieri, M.: Generalized Kähler manifolds, commuting complex structures, and split tangent bundles. Commun. Math. Phys. 271(2), 561–575 (2007)zbMATHGoogle Scholar
  2. 2.
    Bailey, M.: Local classification of generalized complex structures. J. Differ. Geom. 95(1), 1–37 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bailey, M.: Symplectic foliations and generalized complex structures. Can. J. Math. 66(1), 31–56 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bailey, M., Cavalcanti, G., Gualtieri, M.: Type 1 generalized Calabi–Yaus. J. Geom. Phys. 120(Supplement C), 89–95 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brunella, M.: On transversely holomorphic flows I. Invent. Math. 126(2), 265–279 (1996)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cavalcanti, G.R.: New aspects of the \(dd^c\)-lemma. Oxford D.Phil. thesis. (2005) arXiv:math/0501406
  7. 7.
    Cavalcanti, G.R., Gualtieri, M.: A surgery for generalized complex structures on 4-manifolds. J. Diff. Geom. 76, 35–43 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cavalcanti, G.R., Gualtieri, M.: Blow-up of generalized complex 4-manifolds. J. Topol. 2, 840–864 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chu, J., Weinkove, B., Tosatti, V.: The Monge-Ampère equation for non-integrable almost complex structures. J. EMS 21(7), 1949–1984 (2019)zbMATHGoogle Scholar
  10. 10.
    Gates, J., Silvester, J., Hull, C.M., Roc̆ek, M.: Twisted multiplets and new supersymmetric non-linear \(\sigma \)-models. Nucl. Phys. B 248, 157 (1984)Google Scholar
  11. 11.
    Gauduchon, P.: Le théorème de l’excentricité nulle (French). C. R. Acad. Sci. Paris Sér. A-B 285(5), A387–A390 (1977)zbMATHGoogle Scholar
  12. 12.
    Gauduchon, P.: La 1-forme de torsion dune variété hermitienne compacte. Math. Ann. 267, 495–518 (1984)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ghys, Ë.: On transversely holomorphic flows II. Invent. Math. 126(2), 281–286 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Goto, R., Hayano, K.: \(C^\infty \)-logarithmic transformations and generalized complex structures. J. Symplectic Geom. 14(2), 341–357 (2016)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Graña, M., Minasian, R., Petrini, M., Tomasiello, A.: Generalized structures of N = 1 vacua. JHEP 11, 020 (2005)MathSciNetGoogle Scholar
  16. 16.
    Gualtieri, M.: Generalized complex geometry, Oxford D.Phil. thesis. (2004) arXiv:math.DG/0401221
  17. 17.
    Gualtieri, M.: Generalized complex geometry. Ann. Math. (2) 174(1), 75–123 (2011)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gualtieri, M.: Generalized Kahler geometry. Commun. Math. Phys. 331(1), 297–331 (2014)zbMATHGoogle Scholar
  19. 19.
    Hitchin, N.: Generalized Calabi–Yau manifolds. Q. J. Math. 54(3), 281–308 (2003)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Lindström, U., Minasian, R., Tomasiello, A., Zabzine, M.: Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257, 235–256 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lindström, U., Roc̆ek, M., von Unge, R., Zabzine, M.: Generalized Kähler manifolds and off-shell supersymmetry. Commun. Math. Phys. 269, 833–849 (2007)zbMATHGoogle Scholar
  22. 22.
    Matsushita, Y.: Fields of 2-planes and two kinds of almost complex structures on compact 4-dimensional manifolds. Math. Z. 207(2), 281–291 (1991)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Nirenberg, L.: A Complex Frobenius Theorem, Seminar on Analytic Functions, pp. 172–189. Institute for Advanced Study, Princeton (1957)Google Scholar
  24. 24.
    Torres, R.: Constructions of generalized complex structures in dimension four. Commun. Math. Phys. 314(2), 351–371 (2012)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Torres, R., Yazinski, J.: On the number of type change loci of a generalized complex structure. Lett. Math. Phys. 104(4), 451–464 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Wood, J.: Harmonic morphisms, conformal foliations and Seifert fibre spaces. Geometry of low-dimensional manifolds, 1 (Durham). Lond. Math. Soc. Lect. Note Ser. 150, 247–259 (1989)Google Scholar

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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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