Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 10 (2014), 042, 13 pages      arXiv:1404.0505      https://doi.org/10.3842/SIGMA.2014.042
Contribution to the Special Issue on Progress in Twistor Theory

Twistors and Bi-Hermitian Surfaces of Non-Kähler Type

Akira Fujiki a and Massimiliano Pontecorvo b
a) Research Institute for Mathematical Sciences, Kyoto University, Japan
b) Dipartimento di Matematica e Fisica, Università Roma Tre., Italy

Received November 20, 2013, in final form April 04, 2014; Published online April 08, 2014

Abstract
The aim of this work is to give a twistor presentation of recent results about bi-Hermitian metrics on compact complex surfaces with odd first Betti number.

Key words: non-Kähler surfaces; bi-Hermitian metrics; twistor space.

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References

  1. Alekseevsky D.V., Marchiafava S., Pontecorvo M., Compatible complex structures on almost quaternionic manifolds, Trans. Amer. Math. Soc. 351 (1999), 997-1014.
  2. Apostolov V., Bihermitian surfaces with odd first Betti number, Math. Z. 238 (2001), 555-568.
  3. Apostolov V., Bailey M., Dloussky G., From locally conformally Kähler to bi-Hermitian structures on non-Kähler complex surfaces, arXiv:1307.3660.
  4. Apostolov V., Dloussky G., Bihermitian metrics on Hopf surfaces, Math. Res. Lett. 15 (2008), 827-839, arXiv:0710.2266.
  5. Apostolov V., Gauduchon P., Grantcharov G., Bi-Hermitian structures on complex surfaces, Proc. London Math. Soc. 79 (1999), 414-428.
  6. Apostolov V., Gualtieri M., Generalized Kähler manifolds, commuting complex structures, and split tangent bundles, Comm. Math. Phys. 271 (2007), 561-575.
  7. Atiyah M.F., Hitchin N.J., Singer I.M., Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), 425-461.
  8. Bogomolov F.A., Classification of surfaces of class ${\rm VII}_0$ with $b_2=0$, Math. USSR-Izv. 10 (1976), 255-269.
  9. Bombieri E., Letter to Kodaira, 1973.
  10. Boyer C.P., A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), 157-164.
  11. Brunella M., Locally conformally Kähler metrics on Kato surfaces, Nagoya Math. J. 202 (2011), 77-81, arXiv:1001.0530.
  12. Cavalcanti G.R., Gualtieri M., Blowing up generalized Kähler 4-manifolds, Bull. Braz. Math. Soc. (N.S.) 42 (2011), 537-557, arXiv:1106.1481.
  13. Dloussky G., On surfaces of class ${\rm VII}^+_0$ with numerically anticanonical divisor, Amer. J. Math. 128 (2006), 639-670, math.CV/0406387.
  14. Dloussky G., Oeljeklaus K., Vector fields and foliations on compact surfaces of class $\rm VII_0$, Ann. Inst. Fourier (Grenoble) 49 (1999), 1503-1545.
  15. Dloussky G., Oeljeklaus K., Toma M., Class $\rm VII_0$ surfaces with $b_2$ curves, Tohoku Math. J. 55 (2003), 283-309, math.CV/0201010.
  16. Donaldson S., Friedman R., Connected sums of self-dual manifolds and deformations of singular spaces, Nonlinearity 2 (1989), 197-239.
  17. Enoki I., Surfaces of class ${\rm VII}_{0}$ with curves, Tôhoku Math. J. 33 (1981), 453-492.
  18. Fujiki A., Compact self-dual manifolds with torus actions, J. Differential Geom. 55 (2000), 229-324.
  19. Fujiki A., Pontecorvo M., Anti-self-dual bihermitian structures on Inoue surfaces, J. Differential Geom. 85 (2010), 15-71, arXiv:0903.1320.
  20. Gauduchon P., La 1-forme de torsion d'une variété hermitienne compacte, Math. Ann. 267 (1984), 495-518.
  21. Gauduchon P., Ornea L., Locally conformally Kähler metrics on Hopf surfaces, Ann. Inst. Fourier (Grenoble) 48 (1998), 1107-1127.
  22. Gentili G., Salamon S., Stoppato C., Twistor transforms of quaternionic functions and orthogonal complex structures, arXiv:1205.3513.
  23. Goto R., Unobstructed K-deformations of generalized complex structures and bi-Hermitian structures, Adv. Math. 231 (2012), 1041-1067, arXiv:0911.2958.
  24. Griffiths P., Harris J., Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons, New York, 1978.
  25. Gualtieri M., Generalized Kähler geometry, arXiv:1007.3485.
  26. Hausen J., Zur Klassifikation glatter kompakter ${\mathbb C}^*$-Flächen, Math. Ann. 301 (1995), 763-769.
  27. Inoue M., On surfaces of class ${\rm VII}_{0}$, Invent. Math. 24 (1974), 269-310.
  28. Inoue M., New surfaces with no meromorphic functions. II, in Complex Analysis and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 91-106.
  29. Joyce D.D., Explicit construction of self-dual 4-manifolds, Duke Math. J. 77 (1995), 519-552.
  30. Kato M., Compact complex manifolds containing "global" spherical shells. I, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Kinokuniya Book Store, Tokyo, 1978, 45-84.
  31. Kim J., Pontecorvo M., A new method of constructing scalar-flat Kähler surfaces, J. Differential Geom. 41 (1995), 449-477.
  32. LeBrun C., Anti-self-dual Hermitian metrics on blown-up Hopf surfaces, Math. Ann. 289 (1991), 383-392.
  33. Li J., Yau S.-T., Zheng F., On projectively flat Hermitian manifolds, Comm. Anal. Geom. 2 (1994), 103-109.
  34. Lübke M., Teleman A., The Kobayashi-Hitchin correspondence, World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
  35. Nakamura I., On surfaces of class ${\rm VII}_0$ with curves, Invent. Math. 78 (1984), 393-443.
  36. Nakamura I., On surfaces of class ${\rm VII}_0$ with curves. II, Tohoku Math. J. 42 (1990), 475-516.
  37. Pontecorvo M., On twistor spaces of anti-self-dual Hermitian surfaces, Trans. Amer. Math. Soc. 331 (1992), 653-661.
  38. Pontecorvo M., Complex structures on Riemannian four-manifolds, Math. Ann. 309 (1997), 159-177.
  39. Salamon S., Special structures on four-manifolds, Riv. Mat. Univ. Parma (4) 17 (1991), 109-123.
  40. Salamon S., Viaclovsky J., Orthogonal complex structures on domains in ${\mathbb R}^4$, Math. Ann. 343 (2009), 853-899, arXiv:0704.3422.
  41. Teleman A., Projectively flat surfaces and Bogomolov's theorem on class ${\rm VII}_0$ surfaces, Internat. J. Math. 5 (1994), 253-264.
  42. Teleman A., The pseudo-effective cone of a non-Kählerian surface and applications, Math. Ann. 335 (2006), 965-989, arXiv:0704.2948.
  43. Teleman A., Instantons and curves on class VII surfaces, Ann. of Math. 172 (2010), 1749-1804, arXiv:0704.2634.
  44. Teleman A., A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces, arXiv:1309.0350.
  45. Thurston W.P., Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468.


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