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The twistor space of the conformal six sphere and vector bundles on quadrics. (English) Zbl 0856.32020

The Penrose transform establishes a correspondence between the conformal geometry of the standard 4-sphere \(S^4\) and the holomorphic geometry of projective lines on \(\mathbb{P}^3 (\mathbb{C})\). The twistor space construction can be generalized to any even-dimensional, oriented manifold equipped with a conformal structure. Here the author describes the twistor fibration \(\tau : Q_6 \to S^6\) from the smooth quadric hypersurface \(Q_6\) of \(\mathbb{P}^7 (\mathbb{C})\) to the 6-sphere. The reader may enjoy the nice description given here of the geometry of complex planes in \(Q_6\) in terms of Clifford algebras and spinors. The application given here to vector bundles on hyperquadrics, i.e. the triviality of any holomorphic vector bundle \(E\) on a smooth hyperquadric \(Q_t \subset \mathbb{P}^{t + 1}\), \(t \geq 6\), such that \(E |H\) is trivial for some complex plane \(H \subset Q_t\) seems to be known to the specialists (see e.g. the proof of Proposition 2 in the reviewer’s paper Ann. Univ. Ferrara, Nuova Ser., Sez. VII 27, 135-146 (1981; Zbl 0495.14008)] and many related interesting results are contained in papers by G. Ottaviani.
Reviewer: E.Ballico (Povo)

MSC:

32L25 Twistor theory, double fibrations (complex-analytic aspects)
53A30 Conformal differential geometry (MSC2010)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)

Citations:

Zbl 0495.14008
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References:

[1] Atiyah, M. F., Geometry of Yang-Mills Fields (1979), Scuola Normale Superiore: Scuola Normale Superiore Pisa · Zbl 0435.58001
[2] 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 · Zbl 0389.53011
[3] Barth, W., Some properties of stable rank-2 vector bundles on \(P_n\), Math. Ann., 226, 123-150 (1977) · Zbl 0332.32021
[4] Berard, B. L.; Ochiai, T., On some generalisations of the construction of twistor spaces, (Hitchin, N. J.; Willmore, T. J., Global Riemannian Geometry (1984), Ellis Horwood: Ellis Horwood Chichester, UK) · Zbl 0639.53042
[5] Burns, D.; de Bartolomeis, P., Applications harmoniques stables dans \(P_n\), Ann. scient. Ec. Norm. Sup 4ème série, 21, 159-177 (1988) · Zbl 0661.32035
[6] Cartan, E., The Theory of Spinors (1930), Hermann: Hermann Paris · Zbl 0147.40101
[7] Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1978), Wiley: Wiley New York · Zbl 0408.14001
[8] Hitchin, N. J., Linear field equations on self-dual spaces, (Proc. Roy. Soc. London Ser. A, 370 (1980)), 173-191 · Zbl 0436.53058
[9] Inoue, Y., Twistor spaces of even dimensional manifolds, J. Math. Kyoto Univ., 32, 101-134 (1992) · Zbl 0801.53033
[10] Lebrun, C. R., Orthogonal complex structures on \(S^6\), (Proc. Amer. Math. Soc., 101 (1987)), 136-138 · Zbl 0629.53037
[11] O’Brian, N. R.; Rawnsley, J. H., Twistor spaces, Ann. Global Anal. Geom., 3, 29-58 (1985) · Zbl 0526.53057
[12] Slupinski, M. J., Espaces de twisteurs kählériens en dimension \(4k, k > 1\), J. London Math. Soc., 33, 2, 535-542 (1986) · Zbl 0598.53056
[13] Wong, P. M., Twistor spaces over 6-dimensional manifolds, Illinois J. Math., 31, 274-311 (1987) · Zbl 0639.53043
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