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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.
pdf (356 kb)
tex (24 kb)
References
- Alekseevsky D.V., Marchiafava S., Pontecorvo M., Compatible complex structures
on almost quaternionic manifolds, Trans. Amer. Math. Soc.
351 (1999), 997-1014.
- Apostolov V., Bihermitian surfaces with odd first Betti number,
Math. Z. 238 (2001), 555-568.
- Apostolov V., Bailey M., Dloussky G., From locally conformally Kähler to
bi-Hermitian structures on non-Kähler complex surfaces, arXiv:1307.3660.
- Apostolov V., Dloussky G., Bihermitian metrics on Hopf surfaces,
Math. Res. Lett. 15 (2008), 827-839, arXiv:0710.2266.
- Apostolov V., Gauduchon P., Grantcharov G., Bi-Hermitian structures on
complex surfaces, Proc. London Math. Soc. 79 (1999),
414-428.
- Apostolov V., Gualtieri M., Generalized Kähler manifolds, commuting complex
structures, and split tangent bundles, Comm. Math. Phys.
271 (2007), 561-575.
- 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.
- Bogomolov F.A., Classification of surfaces of class ${\rm VII}_0$ with $b_2=0$,
Math. USSR-Izv. 10 (1976), 255-269.
- Bombieri E., Letter to Kodaira, 1973.
- Boyer C.P., A note on hyper-Hermitian four-manifolds, Proc. Amer.
Math. Soc. 102 (1988), 157-164.
- Brunella M., Locally conformally Kähler metrics on Kato surfaces,
Nagoya Math. J. 202 (2011), 77-81, arXiv:1001.0530.
- Cavalcanti G.R., Gualtieri M., Blowing up generalized Kähler 4-manifolds,
Bull. Braz. Math. Soc. (N.S.) 42 (2011), 537-557,
arXiv:1106.1481.
- Dloussky G., On surfaces of class ${\rm VII}^+_0$ with numerically
anticanonical divisor, Amer. J. Math. 128 (2006), 639-670,
math.CV/0406387.
- Dloussky G., Oeljeklaus K., Vector fields and foliations on compact surfaces of
class $\rm VII_0$, Ann. Inst. Fourier (Grenoble) 49
(1999), 1503-1545.
- 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.
- Donaldson S., Friedman R., Connected sums of self-dual manifolds and
deformations of singular spaces, Nonlinearity 2 (1989),
197-239.
- Enoki I., Surfaces of class ${\rm VII}_{0}$ with curves, Tôhoku
Math. J. 33 (1981), 453-492.
- Fujiki A., Compact self-dual manifolds with torus actions,
J. Differential Geom. 55 (2000), 229-324.
- Fujiki A., Pontecorvo M., Anti-self-dual bihermitian structures on Inoue
surfaces, J. Differential Geom. 85 (2010), 15-71,
arXiv:0903.1320.
- Gauduchon P., La 1-forme de torsion d'une variété hermitienne compacte,
Math. Ann. 267 (1984), 495-518.
- Gauduchon P., Ornea L., Locally conformally Kähler metrics on Hopf
surfaces, Ann. Inst. Fourier (Grenoble) 48 (1998),
1107-1127.
- Gentili G., Salamon S., Stoppato C., Twistor transforms of quaternionic
functions and orthogonal complex structures, arXiv:1205.3513.
- Goto R., Unobstructed K-deformations of generalized complex structures and
bi-Hermitian structures, Adv. Math. 231 (2012),
1041-1067, arXiv:0911.2958.
- Griffiths P., Harris J., Principles of algebraic geometry, Pure and Applied Mathematics, John Wiley & Sons,
New York, 1978.
- Gualtieri M., Generalized Kähler geometry, arXiv:1007.3485.
- Hausen J., Zur Klassifikation glatter kompakter ${\mathbb C}^*$-Flächen, Math. Ann. 301 (1995), 763-769.
- Inoue M., On surfaces of class ${\rm VII}_{0}$, Invent. Math.
24 (1974), 269-310.
- Inoue M., New surfaces with no meromorphic functions. II, in Complex Analysis
and Algebraic Geometry, Iwanami Shoten, Tokyo, 1977, 91-106.
- Joyce D.D., Explicit construction of self-dual 4-manifolds, Duke
Math. J. 77 (1995), 519-552.
- 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.
- Kim J., Pontecorvo M., A new method of constructing scalar-flat Kähler
surfaces, J. Differential Geom. 41 (1995), 449-477.
- LeBrun C., Anti-self-dual Hermitian metrics on blown-up Hopf surfaces,
Math. Ann. 289 (1991), 383-392.
- Li J., Yau S.-T., Zheng F., On projectively flat Hermitian manifolds,
Comm. Anal. Geom. 2 (1994), 103-109.
- Lübke M., Teleman A., The Kobayashi-Hitchin correspondence, World
Scientific Publishing Co., Inc., River Edge, NJ, 1995.
- Nakamura I., On surfaces of class ${\rm VII}_0$ with curves, Invent.
Math. 78 (1984), 393-443.
- Nakamura I., On surfaces of class ${\rm VII}_0$ with curves. II,
Tohoku Math. J. 42 (1990), 475-516.
- Pontecorvo M., On twistor spaces of anti-self-dual Hermitian surfaces,
Trans. Amer. Math. Soc. 331 (1992), 653-661.
- Pontecorvo M., Complex structures on Riemannian four-manifolds, Math.
Ann. 309 (1997), 159-177.
- Salamon S., Special structures on four-manifolds, Riv. Mat. Univ.
Parma (4) 17 (1991), 109-123.
- Salamon S., Viaclovsky J., Orthogonal complex structures on domains in
${\mathbb R}^4$, Math. Ann. 343 (2009), 853-899,
arXiv:0704.3422.
- Teleman A., Projectively flat surfaces and Bogomolov's theorem on class
${\rm VII}_0$ surfaces, Internat. J. Math. 5 (1994),
253-264.
- Teleman A., The pseudo-effective cone of a non-Kählerian surface and
applications, Math. Ann. 335 (2006), 965-989,
arXiv:0704.2948.
- Teleman A., Instantons and curves on class VII surfaces, Ann. of
Math. 172 (2010), 1749-1804, arXiv:0704.2634.
- 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.
- Thurston W.P., Some simple examples of symplectic manifolds, Proc.
Amer. Math. Soc. 55 (1976), 467-468.
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