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Twistors, Self-Duality, and Spin${}^c$ Structures
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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The fact that every compact oriented 4-manifold admits spin$^c$ structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and spin$^c$ structures in any dimension.
Keywords: 4-manifold, spin$^c$ structure, twistor space, self-dual 2-form. 
@article{SIGMA_2021_17_a101,
     author = {Claude LeBrun},
     title = {Twistors, {Self-Duality,} and {Spin}${}^c$ {Structures}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a101/}
}
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Claude LeBrun. Twistors, Self-Duality, and Spin${}^c$ Structures. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a101/

[1] 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 | DOI

[2] Gompf R. E., Stipsicz A. I., $4$-manifolds and Kirby calculus, Graduate Studies in Mathematics, 20, Amer. Math. Soc., Providence, RI, 1999 | DOI

[3] Hirzebruch F., Hopf H., “Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten”, Math. Ann., 136 (1958), 156–172 | DOI

[4] Hitchin N. J., “Kählerian twistor spaces”, Proc. London Math. Soc., 43 (1981), 133–150 | DOI

[5] Killingback T. P., Rees E. G., “${\rm Spin}^c$ structures on manifolds”, Classical Quantum Gravity, 2 (1985), 433–438 | DOI

[6] Milnor J. W., Stasheff J. D., Characteristic classes, Annals of Mathematics Studies, 76, Princeton University Press, Princeton, N.J., 1974

[7] Penrose R., “Nonlinear gravitons and curved twistor theory”, Gen. Relativity Gravitation, 7 (1976), 31–52 | DOI

[8] Penrose R., Rindler W., Spinors and space-time, v. 2, Cambridge Monographs on Mathematical Physics, Spinor and twistor methods in space-time geometry, Cambridge University Press, Cambridge, 1986 | DOI

[9] Whitney H., “On the topology of differentiable manifolds”, Lectures in Topology, University of Michigan Press, Ann Arbor, Mich., 1941, 101–141